SAT SUBJECT TEST MATH LEVEL 2
PART 2
REVIEW OF MAJOR TOPICS
CHAPTER 4
Data Analysis, Statistics, and Probability
4.2 Probability
The probability of an event happening is a number defined to be the number of ways the event can happen successfully divided by the total number of ways the event can happen.
EXAMPLES
1. What is the probability of getting a head when a coin is flipped?
A coin can fall in one of two ways, heads or tails. The two are equally likely.
2. What is the probability of getting a 3 when one die is thrown?
A die can fall with any one of six different numbers showing, and there is only one way a 3 can show.
3. What is the probability of getting a sum of 7 when two dice are thrown?
Since it is not obvious how many different throws will produce a sum of 7, or how many different ways the two dice will land, it will be useful to consider all the possible outcomes. The set of all outcomes of an experiment is called the sample space of the experiment. In order to keep track of the elements of the sample space in this experiment, let the first die be green and the second die be red. Since the green die can fall in one of six ways, and the red die can fall in one of six ways, there should be 6 · 6 or 36 elements in the sample space. The elements of the sample space are as follows:
The circled elements of the sample space are those whose sum is 7.
The probability, p, of any event is a number such that 0 p 1. If p = 0, the event cannot happen. If p = 1, the event is sure to happen.
4. (A) What is the probability of getting a 7 when one die is thrown?
(B) What is the probability of getting a number less than 12 when one die is thrown?
SOLUTIONS
(A) P (7) = 0 since a single die has only numbers 1 through 6 on its face.
(B) P (# < 12) = 1 since any face number is less than 12.
The odds in favor of an event happening are defined to be the probability of the event happening divided by the probability of the event not happening .
5. What are the odds in favor of getting a number greater than 2 when one die is thrown?
and . Therefore, the odds in favor of a number greater than or 2:1.
INDEPENDENT EVENTS
Independent events are events that have no effect on one another. Two events are defined to be independent if and only if P() = P (A) · P (B ), where means both events A and B happen. If two events are not independent, they are said to be dependent.
EXAMPLES
1. If two fair coins are flipped, what is the probability of getting two heads?
Since the flip of each coin has no effect on the outcome of any other coin, these are independent events.
2. When two dice are thrown, what is the probability of getting two 5s?
These are independent events because the result of one die does not affect the result of the other.
3. Two dice are thrown. Event A is “the sum of 7.” Event B is “at least one die is a 6.” Are A and B independent?
A = {(1,6), (6,1), (2,5), (5,2), (3,4), (4,3)} and
B = {(1,6), (2,6), (3,6), (4,6), (5,6), (6,6), (6,1), (6,2), (6,3), (6,4), (6,5)}.
Therefore, and . Therefore, .
Therefore, , and so events A and B are dependent.
4. If the probability that John will buy a certain product is , that Bill will buy that product is , and that Sue will buy that product is , what is the probability that at least one of them will buy the product?
Since the purchase by any one of the people does not affect the purchase by anyone else, these events are independent. The best way to approach this problem is to consider the probability that none of them buys the product.
Let A = the event “John does not buy the product.”
Let B = the event “Bill does not buy the product.”
Let C = the event “Sue does not buy the product.”
TIP To find the probability of “at least one,” find 1 – probability of “none.” |
The probability that none of them buys the product . Therefore, the probability that at least one of them buys the product is .
MUTUALLY EXCLUSIVE EVENTS
In general, the probability of event A happening or event B happening or both happening is equal to the sum of P(A) and P(B) less the probability of both happening. In symbols, P(AB) = P(A) + P(B) – P(AB ), where AB means the union of sets A and B. If P(AB) = 0, the events are said to be mutually exclusive.
EXAMPLES
1. What is the probability of drawing a spade or a king from a deck of 52 cards?
Let A = the event “drawing a spade.”
Let B = the event “drawing a king.”
Since there are 13 spades and 4 kings in a deck of cards,
P(AB) = P (drawing the king of spades)
These events are not mutually exclusive.
TIP Generally in probability, “and” means multiply and “or” means add. |
2. In a throw of two dice, what is the probability of getting a sum of 7 or 11?
Let A = the event “throwing a sum of 7.”
Let B = the event “throwing a sum of 11.”
P(AB) = 0, and so these events are mutually exclusive.
P(AB) = P(A) + P(B). From the chart in Example 3,
Therefore,
EXERCISES
1. With the throw of two dice, what is the probability that the sum will be a prime number?
(A)
(B)
(C)
(D)
(E)
2. If a coin is flipped and one die is thrown, what is the probability of getting a head or a 4?
(A)
(B)
(C)
(D)
(E)
3. Three cards are drawn from an ordinary deck of 52 cards. Each card is replaced in the deck before the next card is drawn. What is the probability that at least one of the cards will be a spade?
(A)
(B)
(C)
(D)
(E)
4. A coin is tossed three times. Let A = {three heads occur} and B = {at least one head occurs}. What is P(AB )?
(A)
(B)
(C)
(D)
(E)
5. A class has 12 boys and 4 girls. If three students are selected at random from the class, what is the probability that all will be boys?
(A)
(B)
(C)
(D)
(E)
6. A red box contains eight items, of which three are defective, and a blue box contains five items, of which two are defective. An item is drawn at random from each box. What is the probability that both items will be nondefective?
(A)
(B)
(C)
(D)
(E)
7. A hotel has five single rooms available, for which six men and three women apply. What is the probability that the rooms will be rented to three men and two women?
(A)
(B)
(C)
(D)
(E)
8. Of all the articles in a box, 80% are satisfactory, while 20% are not. The probability of obtaining exactly five good items out of eight randomly selected articles is
(A) 0.003
(B) 0.013
(C) 0.132
(D) 0.147
(E) 0.800
Answers and Explanations
Probability
1. (C) There is 1 way to get a 2, and there are 2 ways to get a 3, 4 ways to get a 5, 6 ways to get a 7, 2 ways to get an 11. Out of 36 elements in the sample space, 15 successes are possible.
2. (D) The probability of getting neither a head nor a 4 is Therefore, probability of getting either is .
3. * (D) Since the drawn cards are replaced, the draws are independent. The probability that none of the cards was a spade Probability that 1 was a spade
4. (E) The only situation when neither of these sets is satisfied occurs when three tails appear.
5. * (D) There are 16 students altogether. The probability that the first person chosen is a boy is . Now there are only 15 students left, of which 11 are boys, so the probability that the second student chosen is also a boy is . By the same reasoning, the probability that the third is a boy is . Therefore, the probability that the first and the second and the third students chosen are all boys is
6. (B) Probability of both items being nondefective
7. * (C) is the number of ways 3 men can be selected. is the number of ways 2 women can be selected. is the total number of ways people can be selected to fill 5 rooms.
8. * (D) Since the problem doesn’t say how many articles are in the box, we must assume that it is an unlimited number. The probability of picking 5 satisfactory items (and therefore 3 unsatisfactory ones) is (0.8)^{5}(0.2)^{3}, and there are ways of doing this. Therefore, the desired probability is (0.8)^{5}(0.2)^{3} ≈ 0.147.