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Chapter 4, Part 1

Repeated ObservationsIndependent Events

The Multiplication RuleConditional Probability

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Repeated Observations / Trials

• In this section, we discuss probabilities that arise when we observe:– The same random phenomenon more than

once.– Two or more different random phenomena,

each of which could effect the others.

• We will use Classical Probability—all outcomes are equally likely.

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Repeated Dice Rolling

• Q: I roll my 20-sided die. What is the probability of rolling a 20?– Ans: 1/20 = 0.05 = 5%

• Q: I roll my die a second time. What is the probability of rolling a 20?– Ans: The same as before: 1/20 = 0.05 = 5%

• Q: I roll my die twice. What is the probability that both rolls are 20?

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Repeated Dice Rolling

• DIFFICULT(?) QUESTIONS:– I roll my die twice. You see that the first roll is

20. What is the probability that the second roll is 20?

– I roll my die twice. You see that the second roll is 20. What is the probability that the first roll was 20?

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Repeated Dice Rolling

• Note the difference between:– Probability that both rolls were 20 (400

possible outcomes).– Probability that one roll was 20, given the

result of the other roll (20 outcomes).

• We usually assume that the two rolls “have no effect on one another,” but we don’t actually know this for certain.

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Selection Without Replacement

• Assume that my hat has 5 pink slips and 5 blue slips.

• I pick one slip, and then pick a second slip without replacing the first.

• Q: What is the probability that the first slip is blue?

• Q: What is the probability that the second slip is blue?

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Selection Without Replacement

• The second probability depends on the result of the first selection.– First slip is Blue: There are 4 Blue, 5 Pink

slips left. The probability that the second slip is Blue is 4/9.

– First slip is Pink: There are 5 Blue, 4 Pink slips left. The probability that the second slip is Blue is 5/9.

• This example illustrates what we will call a conditional probability.

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• The probability of one event occurring, assuming that another event has occurred is called a conditional probability.

• Idea: Knowing/assuming that one event occurred might change the probability of a second event.• This commonly happens when we “select

without replacement.” The first result has an effect on the second (and later) results.

• Note that this didn’t happen with the multiple die rolls. The first result does not have an effect on the second (or so we assume).

Conditional Probability

Conditional Probability

• We use the notation P(B|A), read as “the conditional probability of B, given A.”– Note that the order of events matters!!!

• This means: We assume that event A has occurred, and use this knowledge to compute the (classical) probability of B.

• P(B|A) and P(A|B) will usually be different numbers. Order of events matters!!!

Conditional Probability Example

• I have 5 blue and 5 pink slips in the hat. I choose two slips, without replacing the first.

• P(2nd is blue | 1st is Pink) = ??– Assuming the 1st is pink (note the order of events), the

hat contains 5 blue, 4 pink.– The conditional probability is 5/9.

• P(2nd is blue | 1st is Blue) = ??– After the 1st selection, the hat contains 4 blue, 5 pink.– The conditional probability is 4/9.

The Multiplication Rule

• Notation (only in section 4-4): – We observe a random phenomenon twice.– “A and B” means that event A occurs on the first

observation, and event B occurs on the second observation.

– While this is technically a correct use of “A and B,” it requires that each “outcome” consist of two separate observations. This will probably confuse you, so…

– I will use the notation “A then B” to emphasize that we are considering two observations.

The Multiplication Rule

• For any two events A, B.

P(A then B) = P(A) x P(B|A)

• Example: I choose two slips from the hat (without replacing the first). The probability that both slips are blue is:

P(Blue then Blue) = (5/10)*(4/9) = 2/9.

The Multiplication Rule

• P(A then B) = P(A) x P(B|A)

• The probability that the 1st is blue and the 2nd is pink?

P(Blue then Pink) = (5/10)*(5/9) = 5/18.

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The Actual Definition of Conditional Probability

P(A)

B) andP(A A)|P(B

• The conditional probability of B given A (in that order) is defined to be:

• This is the standard use of “A and B” discussed last time. “A then B” is just a special case of this.

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Classical Conditional Probability

• For conditional probability, we assume that one event (A) occurred, and use this information to compute the probability of a second event (B).

• If we make a single observation, and all outcomes are equally likely, there is a very simple method for computing conditional probability…

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Classical Conditional Probability

• For the result of a single observation, P(B|A) is given by:

# of outcomes in “A and B”

# of outcomes in A

• In other words, we take the number of outcomes satisfying BOTH conditions, and divide by the number of outcomes satisfying the GIVEN condition.

Examples

• I select a student from the class. Assume all outcomes are equally likely, compute:– P(Female | Row 1)– P(Male | Back Row)– P(Back Row | Male)– P(Back Row)

Dependent Events

• More often than not, knowing that one event occurred will change our expectations about a second event.

• In terms of conditional probability, this means that P(B) ≠ P(B|A).

• If this is the case, we say that the two events are dependent.

Independent Events

• But in some cases, knowing that one event occurred DOES NOT change our expectations about a second event.

• In terms of conditional probability, this means that P(B) = P(B|A). If this is true, we say the events are independent.

• An example is the 20-sided die:

P(2nd is 20) = P(2nd is 20 | 1st is 20).

Independent Events, Example

• I have 5 blue, 5 pink slips in the hat. I choose two slips, REPLACING THE FIRST before drawing the second. Compute the following:– P(2nd is Blue)– P(2nd is Blue | 1st is Blue)

• You should find these to be the same number, so these are independent events.

The Multiplication Rule for Independent Events

• Original multiplication rule:

P(A then B) = P(A) x P(B|A)

• If the events are independent, then P(B|A) is equal to P(B). The multiplication rule reduces to:

P(A then B) = P(A) x P(B)