Probability
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Probability Overview In this section we will cover: Expressing probabilities GMAT Probability Rule #1 GMAT Probability Rule #2 GMAT Probability Rule #3 GMAT Probability Rule #4
Probability You may encounter questions on the GMAT that ask the probability of an event occuring. For instance: •
Sara rolls two fair, six-sided dice. What is the probability she will roll two 6s?
•
Roland flip two fair coins. What is the probability that neither of the coins will land on tails?
•
A bag contains six blue marbles and six red marbles. If Teresa randomly chooses a marble from the bag, what is the probability that the marble is blue?
All of these problems are probability problems. Probability is a measure of the likelihood of an event occurring.
Expressing Probability The probability of an event occurring is expressed as a number between 0 and 1. A probability of 0 means the event can never happen. A probability of 1 means the event is certain to happen. You may see probability express on the GMAT in terms of a percent, a decimal, or a fraction. For instance, the probability of a fair coin landing on tails can be expressed as
Fraction 1/2 Decimal .5 Percent 50%
Sometimes you may see notation similar to that below: P(A) = 1/2 This means that the probability of an event A is 1/2.
Probability Rules There are four probability rules you need to memorize in order to master GMAT probability questions: 1. The probability of an event A occurring is the number of outcomes that result in A divided by the total number of possible outcomes. 2. The probability of an event occurring plus the probability of the event not occurring equals 1. 3. The probability of event A AND event B occurring is the probability of event A times the probability of event B given that event A has already occurred. 4. The probability of event A OR event B occurring is the probability of event A occurring plus the probability of event B occurring minus the probability of both events occurring. The probability of an event A occurring is the number of outcomes that result in A divided by the total number of possible outcomes. Example: Raphael tosses a fair coin. What is the probability the coin will come up heads? Probability of heads = [heads]/[heads, tails] Probability of heads = 1/2 Example: Tom rolls a fair die. What is the probability that the die will roll an even number? Probability of even number = [2, 4, 6]/[1, 2, 3, 4, 5, 6] Probability of even number = 3/6 Probability of even number = ½
Rule #2 The probability of an event occurring plus the probability of the event not occurring equals 1. In other words, we can say with 100% certainty that an event will either occur or not occur. For instance, the probability of a fair, six-sided die rolling a 4 is 1/6. The probability of the die not landing on 4 is (1 - 1/6) or 5/6. 1/6 + 5/6 = 1. This concept can be very helpful on the GMAT. Sometimes it is easier to determine the probability of an event not occurring than determining the probability of an event occurring. Once your know the probability of an event not occurring, you can subtract the probability from 1 to find the probability of an event occurring.
Rule #3
The probability of event A AND event B occurring is the probability of event A times the probability of event B given that event A has already occurred. Example: Joseph rolls two fair, six-sided die. What is the probability that both die will roll a 6? Probability Probability Probability Probability
of of of of
1st die coming up 6: 1/6 2nd die coming up 6: 1/6 both die coming up 6: (1/6) * (1/6) both die coming up 6: 1/36
Example: A bag contains three blue marbles and three red marbles. If two marbles are drawn randomly from the bag, what is the probability that they are both blue? This problem is a dependent probability. Two events are said to be dependent events if the outcome of one event affects the outcome of the other event. The probability of drawing the second marble depends on the outcome of the first marble. If the first marble is red, there is no possibility of drawing two blue marbles. Thus, the probability of drawing a second blue marble is calculated after the first blue marble has been drawn. Probability of drawing blue on first draw: 3/6 If a blue is drawn on the first draw, there are three red marbles and two blue marbles remaining in the bag. Probability of drawing blue on second draw (given that first was blue): 2/5 Probability of drawing two blue: (3/6) * (2/5) Probability of drawing two blue: 6/30 Probability of drawing two blue: 1/5
Rule #4 The probability of event A OR event B occurring is the probability of event A occurring plus the probability of event B occurring minus the probability of both events occurring. Example: Charles rolls a fair, six-sided die. What is the probability of Charles rolling a 2 or a 4? Probability Probability Probability Probability Probability
of of of of of
2: 1/6 4: 1/6 a 2 or 4: 1/6 + 1/6 a 2 or 4: 2/6 a 2 or 4: 1/3
In the previous problem, the events were mutually exclusive. Mutually exclusive means that the events cannot occur together. There is no way to roll a 2 and a 4 at the same time. The events in the following problem are NOT mutually exclusive.
Example: Of the 100 students at a certain school, 30 students are taking a chemistry class, 40 students are taking a physics class, and 20 students are taking both a physics and a chemistry class. If a student is chosen at random from the school, what is the probability that he or she is taking a physics or a chemistry class? Probability of selecting a student taking a chemistry course: 30/100 Probability of selecting a student taking a physics class: 40/100 Probability of selecting a student taking both classes: 20/100 Probability of a selecting a student taking chemistry OR a student taking physics: 30/100 + 40/100 - 20/100 = 50/100 or 1/2 There are 9 beads in a bag. 3 beads are red, 3 beads are blue, and 3 beads are black. If two beads are chosen at random, what is the probability that they are both blue? A. 1/81 B. 1/12 C. 2/9 D. 1/3 E. 1/4 A letter is randomly selected from the word Mississippi. What is the probability that the letter will be an s? A. 1/11 B. 3/10 C. 4/11 D. 1/4 E. 1/3 A certain deck of cards contains 2 blue cards, 2 red cards, 2 yellow cards, and 2 green cards. If two cards are randomly drawn from the deck, what is the probability that they will both are not blue? A. 15/28 B. 1/4 C. 9/16 D. 1/32 E. 1/16 A fair coin is tossed, and a fair six-sided die is rolled. What is the probability that the coin come up heads and the die will come up 1 or 2? A. 1/2 B. 1/4 C. 1/6 D. 1/12
E. 1/3
A bag of 10 marbles contains 3 red marbles and 7 blue marbles. If two marbles are selected at random, what is the probability that at least one marble is blue? A. 21/50 B. 3/13 C. 47/50 D. 14/15 E. 1/5 A fair, six-sided die is rolled. What is the probability that the number will be odd? A. 1/4 B. 1/2 C. 1/3 D. 1/6 E. 1/5 A letter is randomly select from the word studious. What is the probability that the letter be a U? A. 1/8 B. 1/4 C. 1/3 D. 1/2 E. 3/8 A bag contains 2 red beads, 2 blue beads, and 2 green beads. Sara randomly draws a bead from the bag, and then Victor randomly draws a bead from the bag. What is the probability that Sara will draw a red marble and Victor will draw a blue marble? A. 2/13 B. 1/5 C. 1/3 D. 1/10 E. 2/15 If two fair, six-sided dice are rolled, what is the probability that the sum of the numbers will be 5? A. 1/6
B. 1/4 C. 1/36 D. 1/18 E. 1/9 If four fair coins are tossed, what is the probability of all four coming up heads? A. 1/4 B. 1/6 C. 1/8 D. 1/16 E. 1/32 The probability that a certain event will occur is 5/9. What is the probability that the event will NOT occur? A. 5/9 B. 4/9 C. 2/9 D. 1/4 E. 1/2 A certain bag contains red, blue, yellow, and green marbles. If a marble is randomly drawn from the bag, the probability of drawing a blue marble is .2, the probability of drawing a red marble is .3, and the probability of drawing a yellow marble is .1. What is the probability of drawing a green marble? A. .5 B. .6 C. .2 D. .4 E. .3 A bag contains 3 red marbles, 3 blue marbles, and 3 green marbles. If a marble is randomly drawn from the bag and a fair, six-sided die is tossed, what is the probability of obtaining a red marble and a 6? A. 1/15 B. 1/6 C. 1/3 D. 1/4 E. 1/18 A fair, six-sided die is rolled. What is the probability of obtaining a 3 or an odd number?
A. 1/6 B. 1/5 C. 1/4 D. 2/3 E. 1/2 At a certain business school, 400 students are members of the sailing club, the wine club, or both. If 200 students are members of the wine club and 50 students are members of both clubs, what is the probability that a student chosen at random is a member of the sailing club? A. 1/2 B. 5/8 C. 1/4 D. 3/8 E. 3/5 A bag contains six marbles: two red, two blue, and two green. If two marbles are drawn at random, what is the probability that they are the same color? A. 1/3 B. 1/2 C. 1/8 D. 1/4 E. 1/5 There are five students in a study group: two finance majors and three accounting majors. If two students are chosen at random, what is the probability that they are both accounting students? A. 3/10 B. 2/5 C. 1/5 D. 3/5 E. 4/5 Seven beads are in a bag: three blue, two red, and two green. If three beads are randomly drawn from the bag, what is the probability that they are not all blue? A. 5/7 B. 23/24 C. 6/7 D. 34/35 E. 8/13
A bag has six red marbles and six blue marbles. If two marbles are drawn randomly from the bag, what is the probability that they will both be red? A. 1/2 B. 11/12 C. 5/12 D. 5/22 E. 1/3
Probability You may encounter questions on the GMAT that ask the probability of an event occuring. For instance: •
Sara rolls two fair, six-sided dice. What is the probability she will roll two 6s?
•
Roland flip two fair coins. What is the probability that neither of the coins will land on tails?
•
A bag contains six blue marbles and six red marbles. If Teresa randomly chooses a marble from the bag, what is the probability that the marble is blue?
All of these problems are probability problems. Probability is a measure of the likelihood of an event occurring.
Expressing Probability The probability of an event occurring is expressed as a number between 0 and 1. A probability of 0 means the event can never happen. A probability of 1 means the event is certain to happen. You may see probability express on the GMAT in terms of a percent, a decimal, or a fraction. For instance, the probability of a fair coin landing on tails can be expressed as
Fraction 1/2 Decimal .5 Percent 50%
Sometimes you may see notation similar to that below: P(A) = 1/2 This means that the probability of an event A is 1/2.
Probability Rules There are four probability rules you need to memorize in order to master GMAT probability questions: 1. The probability of an event A occurring is the number of outcomes that result in A divided by the total number of possible outcomes. 2. The probability of an event occurring plus the probability of the event not occurring equals 1. 3. The probability of event A AND event B occurring is the probability of event A times the probability of event B given that event A has already occurred. 4. The probability of event A OR event B occurring is the probability of event A occurring plus the probability of event B occurring minus the probability of both events occurring. The probability of an event A occurring is the number of outcomes that result in A divided by the total number of possible outcomes. Example: Raphael tosses a fair coin. What is the probability the coin will come up heads? Probability of heads = [heads]/[heads, tails] Probability of heads = 1/2 Example: Tom rolls a fair die. What is the probability that the die will roll an even number? Probability of even number = [2, 4, 6]/[1, 2, 3, 4, 5, 6] Probability of even number = 3/6 Probability of even number = ½
Rule #2 The probability of an event occurring plus the probability of the event not occurring equals 1. In other words, we can say with 100% certainty that an event will either occur or not occur. For instance, the probability of a fair, six-sided die rolling a 4 is 1/6. The probability of the die not landing on 4 is (1 - 1/6) or 5/6. 1/6 + 5/6 = 1. This concept can be very helpful on the GMAT. Sometimes it is easier to determine the probability of an event not occurring than determining the probability of an event occurring. Once your know the probability of an event not occurring, you can subtract the probability from 1 to find the probability of an event occurring.
Rule #3
The probability of event A AND event B occurring is the probability of event A times the probability of event B given that event A has already occurred. Example: Joseph rolls two fair, six-sided die. What is the probability that both die will roll a 6? Probability Probability Probability Probability
of of of of
1st die coming up 6: 1/6 2nd die coming up 6: 1/6 both die coming up 6: (1/6) * (1/6) both die coming up 6: 1/36
Example: A bag contains three blue marbles and three red marbles. If two marbles are drawn randomly from the bag, what is the probability that they are both blue? This problem is a dependent probability. Two events are said to be dependent events if the outcome of one event affects the outcome of the other event. The probability of drawing the second marble depends on the outcome of the first marble. If the first marble is red, there is no possibility of drawing two blue marbles. Thus, the probability of drawing a second blue marble is calculated after the first blue marble has been drawn. Probability of drawing blue on first draw: 3/6 If a blue is drawn on the first draw, there are three red marbles and two blue marbles remaining in the bag. Probability of drawing blue on second draw (given that first was blue): 2/5 Probability of drawing two blue: (3/6) * (2/5) Probability of drawing two blue: 6/30 Probability of drawing two blue: 1/5
Rule #4 The probability of event A OR event B occurring is the probability of event A occurring plus the probability of event B occurring minus the probability of both events occurring. Example: Charles rolls a fair, six-sided die. What is the probability of Charles rolling a 2 or a 4? Probability Probability Probability Probability Probability
of of of of of
2: 1/6 4: 1/6 a 2 or 4: 1/6 + 1/6 a 2 or 4: 2/6 a 2 or 4: 1/3
In the previous problem, the events were mutually exclusive. Mutually exclusive means that the events cannot occur together. There is no way to roll a 2 and a 4 at the same time. The events in the following problem are NOT mutually exclusive.
Example: Of the 100 students at a certain school, 30 students are taking a chemistry class, 40 students are taking a physics class, and 20 students are taking both a physics and a chemistry class. If a student is chosen at random from the school, what is the probability that he or she is taking a physics or a chemistry class? Probability of selecting a student taking a chemistry course: 30/100 Probability of selecting a student taking a physics class: 40/100 Probability of selecting a student taking both classes: 20/100 Probability of a selecting a student taking chemistry OR a student taking physics: 30/100 + 40/100 - 20/100 = 50/100 or 1/2 There are 9 beads in a bag. 3 beads are red, 3 beads are blue, and 3 beads are black. If two beads are chosen at random, what is the probability that they are both blue? A. 1/81 B. 1/12 C. 2/9 D. 1/3 E. 1/4 A letter is randomly selected from the word Mississippi. What is the probability that the letter will be an s? A. 1/11 B. 3/10 C. 4/11 D. 1/4 E. 1/3 A certain deck of cards contains 2 blue cards, 2 red cards, 2 yellow cards, and 2 green cards. If two cards are randomly drawn from the deck, what is the probability that they will both are not blue? A. 15/28 B. 1/4 C. 9/16 D. 1/32 E. 1/16 A fair coin is tossed, and a fair six-sided die is rolled. What is the probability that the coin come up heads and the die will come up 1 or 2? A. 1/2 B. 1/4 C. 1/6 D. 1/12
E. 1/3
A bag of 10 marbles contains 3 red marbles and 7 blue marbles. If two marbles are selected at random, what is the probability that at least one marble is blue? A. 21/50 B. 3/13 C. 47/50 D. 14/15 E. 1/5 A fair, six-sided die is rolled. What is the probability that the number will be odd? A. 1/4 B. 1/2 C. 1/3 D. 1/6 E. 1/5 A letter is randomly select from the word studious. What is the probability that the letter be a U? A. 1/8 B. 1/4 C. 1/3 D. 1/2 E. 3/8 A bag contains 2 red beads, 2 blue beads, and 2 green beads. Sara randomly draws a bead from the bag, and then Victor randomly draws a bead from the bag. What is the probability that Sara will draw a red marble and Victor will draw a blue marble? A. 2/13 B. 1/5 C. 1/3 D. 1/10 E. 2/15 If two fair, six-sided dice are rolled, what is the probability that the sum of the numbers will be 5? A. 1/6
B. 1/4 C. 1/36 D. 1/18 E. 1/9 If four fair coins are tossed, what is the probability of all four coming up heads? A. 1/4 B. 1/6 C. 1/8 D. 1/16 E. 1/32 The probability that a certain event will occur is 5/9. What is the probability that the event will NOT occur? A. 5/9 B. 4/9 C. 2/9 D. 1/4 E. 1/2 A certain bag contains red, blue, yellow, and green marbles. If a marble is randomly drawn from the bag, the probability of drawing a blue marble is .2, the probability of drawing a red marble is .3, and the probability of drawing a yellow marble is .1. What is the probability of drawing a green marble? A. .5 B. .6 C. .2 D. .4 E. .3 A bag contains 3 red marbles, 3 blue marbles, and 3 green marbles. If a marble is randomly drawn from the bag and a fair, six-sided die is tossed, what is the probability of obtaining a red marble and a 6? A. 1/15 B. 1/6 C. 1/3 D. 1/4 E. 1/18 A fair, six-sided die is rolled. What is the probability of obtaining a 3 or an odd number?
A. 1/6 B. 1/5 C. 1/4 D. 2/3 E. 1/2 At a certain business school, 400 students are members of the sailing club, the wine club, or both. If 200 students are members of the wine club and 50 students are members of both clubs, what is the probability that a student chosen at random is a member of the sailing club? A. 1/2 B. 5/8 C. 1/4 D. 3/8 E. 3/5 A bag contains six marbles: two red, two blue, and two green. If two marbles are drawn at random, what is the probability that they are the same color? A. 1/3 B. 1/2 C. 1/8 D. 1/4 E. 1/5 There are five students in a study group: two finance majors and three accounting majors. If two students are chosen at random, what is the probability that they are both accounting students? A. 3/10 B. 2/5 C. 1/5 D. 3/5 E. 4/5 Seven beads are in a bag: three blue, two red, and two green. If three beads are randomly drawn from the bag, what is the probability that they are not all blue? A. 5/7 B. 23/24 C. 6/7 D. 34/35 E. 8/13
A bag has six red marbles and six blue marbles. If two marbles are drawn randomly from the bag, what is the probability that they will both be red? A. 1/2 B. 11/12 C. 5/12 D. 5/22 E. 1/3
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