1. Probability and Statistics Chapter 3 Notes
Section 3-2
I. Conditional Probability
A conditional probability is the probability of an event occurring, given that another event has already occurred.
The conditional probability of event B occurring, given that event A has occurred, is denoted 𝑃 𝐵 𝐴 and is read as “probability of B, given A.”
II. Independent and Dependent Events
Two events are independent if the occurrence of one of the events does not affect the probability of the occurrence of the other event.
Two events A and B are independent if 𝑃 𝐵 𝐴 =P(B), or if 𝑃 𝐴 𝐵 =P(A).
In other words, event B is equally likely to occur whether event A has occurred or not.
Events that are not independent are dependent.
To determine if A and B are independent, first calculate P(B), the probability of event B. Then calculate 𝑃 𝐵 𝐴 , the probability of B, given A.
If the two probabilities are equal, the events are independent.
If the two probabilities are not equal, the events are dependent.

2. Probability and Statistics Chapter 3 Notes
Section 3-2
III. The Multiplication Rule
P(A and B)=P(A)* 𝑃 𝐵 𝐴
If Events A and B are dependent;
1) Find the probability the first event occurs
2) Find the probability the second event occurs given the first event has occurred.
3) Multiply these two probabilities to find the probability that both events will occur in sequence.
If Events A and B are independent, P(A and B)=P(A)*P(B).
This simplified rule can be extended for any number of independent events, just like the Fundamental Counting Principle could be extended.

3. Example 1 (Page 149)
1) Two cards are selected in sequence from a standard deck. Find the probability that the second card is a queen, given that the first card is a king. Assume that the king is not replaced.
2) The table shows the results of a study in which researchers examined a child’s IQ and the presence of a specific gene in the child. Find the probability that a child has a high IQ given that the child has the gene.
SOLUTIONS:
1) The probability of drawing a queen after a king (or any other card) has been taken out of the deck is , or about
2) There are 72 children who have the gene. They are our sample space. Of these, 33 have a high IQ.
So, 𝑃 𝐵 𝐴 = ≈0.458
Gene Present
Gene Not Present
Total
High IQ 33 19 52
Normal IQ 39 11 50 72 30
102

4. Example 2 (Page 150)
Decide whether the events are independent or dependent.
1) Selecting a king from a standard deck (A), not replacing it, and then selecting a queen from the deck (B).
2) Tossing a coin and getting a head (A), and then rolling a six-sided die and obtaining a 6 (B).
SOLUTIONS:
1) The probability of pulling a queen out of the deck is
The probability of pulling a queen out of the deck after a king has already been removed is
Since these probabilities are different, the events are dependent.
2) The probability of obtaining a 6 is
The probability of obtaining a 6 given that the coin came up heads is
Since these probabilities are equal, the events are independent.

5. Example 3 (Page 151)
1) Two cards are selected, without replacing the first card, from a standard deck. Find the probability of selecting a king and then selecting a queen.
2) A coin is tossed and a die is rolled. Find the probability of getting a head and then rolling a 6.
SOLUTIONS:
1) Because the first card is not replaced, the events are dependent.
P(K and Q)=P(K)* 𝑃 𝑄 𝐾
= * = ≈ 0.006
2) The two events are independent.
P(H and 6) =P(H)*P(6)
= 1 2 * = ≈ 0.083

6. Example 4 (Page 152)
1) A coin is tossed and a die is rolled. Find the probability of getting a tail and then rolling a 2.
2) The probability that a particular knee surgery is successful is Find the probability that three surgeries in a row are all successful.
3) Find the probability that none of the three knee surgeries is successful.
4) Find the probability that at least one of the three knee surgeries is successful.
SOLUTIONS
1) The probability of getting a tail and then rolling a 2 is
P(T and 2)=P(T)*P(2)
= 1 2 * = ≈ 0.083

7. 2). The probability that each knee surgery is successful is 0. 85
2) The probability that each knee surgery is successful is Since these events are independent, the probability that all three are successful is found by multiplying their probabilities together.
(.85)(.85)(.85) ≈ 0.614
The probability that all three surgeries are successful is about
3) Because the probability of success for one surgery is .85, the probability of failure for one surgery is 1-.85=.15
This is true because failure is the complement of success.
The probability that each knee surgery is not successful is Since these events are independent, the probability that all three are not successful is found by multiplying their probabilities together.
(.15)(.15)(.15) ≈ 0.003
The probability that none of the surgeries are successful is about

8. 4). The phrase “at least one” means one or more
4) The phrase “at least one” means one or more. The complement to the event “at least one is successful” is the event “none are successful”.
Using the complement rule, we can simply subtract the probability that none were successful from 1 to find the probability that at least one was successful.
= 0.997
The probability that at least one of the surgeries is successful is about

9. Example 5 (Page 153)
More than 15,000 medical school seniors applied to residency programs in Of those, 93% were matched to a residency position. 74% of the seniors matched to a residency position were matched to one of their top two choices.
1) Find the probability that a randomly selected senior was matched to a residency program and it was one of the senior’s top two choices.
2) Find the probability that a randomly selected senior that was matched to a residency program did not get matched with one of the senior’s top two choices.
3) Would it be unusual for a randomly selected senior to result in a senior that was matched to a residency position and it was one of the senior’s top two choices?
SOLUTION:
1) These two events are dependent:
P(A and B) = P(A)* 𝑃 𝐵 𝐴 =(0.93)(0.74) ≈ 0.688
2) To find this probability, use the complement:
𝑃 𝐵′ 𝐴 =1−𝑃 𝐵 𝐴 =1−0.74=0.26
3) No, this event occurs around 68.8% of the time.

10. Assignments:
Classwork: Pages #1-12 All
Homework: Pages #14-30 Evens