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AP Statistics Notes: 6.2 – Rules of Probability

Objective(s): The learner will be able to use Venn diagrams and two-way tables and the rules of probability to find probabilites of outcomes.

Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run.

Random- individual outcomes are uncertain, but there is a regular distribution of outcomes in a large number of repetitions

1. Randomness requires independent trials

2. Probability is empirical - have to observe it

Probability Rules:

1) 0 ≤ P(x) ≤ 1

2) P(S) = 1

Definitions:

outcome – a result that occurs (“head”, “4 and 2”)

sample space – a set of all possible outcomes (“flip a coin - H, T”, “roll a die – 1, 2, 3, 4, 5, 6”)

event – a particular set of outcomes (“even”, “6”, “H”)

union (A [pic] B) - all outcomes in A or B (or both); P(A [pic]B) = P(A) + P(B) - P(A [pic] B)

intersection (A ∩ B ) – all outcomes in A and B

disjoint: no common outcomes; P(A [pic]B) = 0

independent events: events whose outcomes don’t affect each other

if A and B are independent: P(A [pic]B) = P(A) x P(B)

complement (Ac): the probability that an event doesn’t happen (“not a tail”, “not a sum of 7”)

P(Ac) = 1 – P(A)

EXAMPLE #1 – List the sample space if you roll two dice.

EXAMPLE #2 – List the sample space for rolling a die and flipping a coin.

tree diagram – summarizes possible outcomes with multiple events using branches to visualize

multiplication principle - if one task can be done n1 ways and a second task can be done n2 ways, both tasks can be done in n1 x n2 number of ways

EXAMPLE #3 – Use a tree diagram to list the possible outcomes if you flip a coin three times. Write the sample space for “number of heads” and find the probabilities for each outcome.

EXAMPLE #4 – Find P(5 heads on 5 tosses)

EXAMPLE #5 – Find the probability of answering the last three MC questions correctly using only random guessing if all questions have 5 choices.

EXAMPLE #6 – Find the probability of getting at least one girl in seven births.

EXAMPLE #7 – find the probability of drawing 2 hearts from a deck of cards at the same time.

EXAMPLE #8 – One on One Basketball: If you make one, you get another shot. P(make a shot) = .6

a) P(0) = b) P(1) = c) P(2) =

EXAMPLE #9 - Let A be the event that a person has red hair. Let B be the event that a person has blue eyes. If P(A) = 0.2, P(B) = 0.4, and P(A and B) = 0.15, make a Venn diagram and a data summary table. Describe the following events in words and find their probabilities:

a. Ac

b. Bc

c. A [pic] B

d. A [pic]B

e. Ac [pic] B

f. Ac [pic] Bc

Homework: p411 23, 24 p 416 30, 32, 33, 35, 36 p423 38 – 41, 43, 44 p430 45 - 52

AP Statistics Notes: 6.2 – Rules of Probability Practice

HINT: Find P(A [pic]B) first and then complete the table or diagram.

1. A is the event that a person is female. B is the event that a person likes math. If P(A) = 0.6, P(B) = 0.2, and P(A [pic]B) = 0.1, describe the following events in words and find their probabilities (using a table or a Venn diagram).

a. Ac b. Bc

c. A [pic] B d. A [pic]B

e. Ac [pic] B f. Ac [pic] Bc

2. A is the event someone likes soccer. B is the event that someone likes reading. If P(A) = 0.2, P(B) = 0.3, and P(A [pic] B) = 0.4, describe the following events in words and find their probabilities (using a table or a Venn diagram).

a. Ac

b. Bc

c. A [pic] B

d. A [pic]B

e. Ac [pic] B

f. Ac [pic] Bc

3. A is the event someone likes pizza. B is the event that a person is male. If P(A) = 0.6, P(B) = 0.3, and A and B are disjoint events, describe the following events in words and find their probabilities (using a table or a Venn diagram).

a. Ac

b. Bc

c. A [pic] B

d. A [pic]B

e. Ac [pic] B

f. Ac [pic] Bc

4. A is the event that a person is female. B is the event that a person likes math. If P(A) = 0.5, P(Bc) = 0.8, and P(A [pic]B) = 0.05, find the probabilities (using a table or a Venn diagram):

a. Ac b. Bc

c. A [pic] B d. A [pic]B

e. Ac [pic] B f. Ac [pic] B

5. A is the event someone likes soccer. B is the event that someone likes reading. If P(Ac) = 0.3, P(Bc) = 0.5, and P(A [pic]B) = 0.90, find the probabilities (using a table or a Venn diagram):

a. Ac

b. Bc

c. A [pic] B

d. A [pic]B

e. Ac [pic] B

f. Ac [pic] Bc

6. A is the event someone likes pizza. B is the event that a person is male. If P(A) = 0.5, P(B) = 0.5, and P(A [pic]Bc) = 0.15, find the probabilities (using a table or a Venn diagram):

a. Ac

b. Bc

c. A [pic] B

d. A [pic]B

e. Ac [pic] B

f. A [pic] Bc

7) If the probability that a modeling agency will get a contract with a designer is .3, what is the probability that it will not win the contract?

8) If the probabilities that Ronny, Johnny, and Donny will be selected as class clown are .5, .3, and .2, respectively, what is the probability that Ronny or Donny will be class clown?

9) If a psychic in an extrasensory perception experiment has called heads or tails correctly on ten successive tosses of a coin, what is the probability that guessing would have yielded this perfect score?

10) Suppose three branches of a local bank average, respectively 120, 180, and 100 clients per day. Suppose further that the probabilities that a client will transact business involving more than $100 during a visit are, respectively, .5, .6, and .7. A client is chosen at random. What is the probability that the client will transact business involving over $100? What is the probability that the client went to the first branch given that she transacted business involving $100?

11) Morris’s kidneys have failed and he is awaiting a kidney transplant. His doctor gives him this information for patients in his condition: 90% survive the transplant, and 10% die. The transplant succeeds in 60% of those who survive, and the other 40% must return to kidney dialysis. The proportions who survive five years are 70% for those with new kidneys and 50% for those who return to dialysis. Morris wants to know the probability that he will survive for 5 years.

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