Intro: Yesterday we talked about “disjoint” events - events where there was no overlap in the outcomes. Rule 4 told us:

P(A or B) = P(A) + P(B).

This answers the question, “What’s the probability that A or B happens?”
Now we want to answer the question, “What’s the probability that A and B happen?”

Example: What’s the probability that when we flip a coin twice, we get two consecutive heads?
Solution: We’re really trying to find the P(head and head). Experience tells us that the probability of getting a head on the first toss is 0.5 and the probability of getting a head on the second toss is 0.5. So together it’s 0.5 • 0.5 = 0.25 or ¹/₄.
The key point to remember is that getting a head on the first toss does not alter our odds of getting a head on the second toss. This is independence.

Venn Diagram:



Independent Events: The occurrence of one event does not alter the probability of the occurrence of another event.

Example: Flipping a coin = indep. events
Example: Drawing a card from a deck = dependent events.
Multiplication Rule for Independent events

P(A and B) = P(A) • P(B)

Note: If you have three independent events, the formula changes logically:

P(A and B and C) = P(A) • P(B) • P(C)

Your turn to dazzle me:
1. Does rolling a six-sided die yield independent or dependent events?
2. Find the P(6) on one toss.
3. Find the P(37) on one toss.
4. Find the P(3 and 4)
5. Find the probability of rolling seven consecutive “2s.”
6. On the TV show, “Who wants to be pestered by Regis?” contestants answer 15 questions that each have four possible responses. Assume that each question is an independent event. Also, assume that the contestant is a New York Mets fan, and thus, totally clueless as to what the answers are.
a. What’s the probability of getting the first question right without using a lifeline?

b. What’s the probability of getting all fifteen questions right without a lifeline?

7. Suppose that they’re introducing a new version of the game show whereby the goal is to get the answer wrong. Again, assume each question is an independent event and that the contestant is still clueless.
a. What’s the probability of getting the first question wrong?
b. What’s the probability of getting all fifteen questions wrong?

8. Let A be the event {getting the question right}. What is A^c?
9. What’s the sample space for each game show question?
10. What’s P(S), if S = sample space?
11. A diagnostic test for the presence of AIDS has a probability of 0.005 of producing a false positive - saying you have AIDS when you don’t. Suppose 140 hospital employees are tested and all 140 are free of AIDS. What is the probability that at least one false positive will occur? (Assume independence)
Solution: Key phrase = at least
P(at least one positive)
= 1 - P(no positives) Complement Rule
= 1 - P(140 negatives)
= 1 - 0.995¹⁴⁰ Multiplication Rule
= 1-0.496
= 0.504.
Conclusion: The probability is greater than ¹/₂ that at least one of the 140 employees will test positive for AIDS, even though no one has the virus!
HW: pg 306 #4.18-4.36 (DUE Nov 6th)
Quiz Wed November 7th, 4.1-4.2