Probability Class 33 1. Homework Check Assignment: Chapter 7 – Exercise 7.68, 7.69 and 7.70 Reading: Chapter 7 – p. 238-247 2.

Probability

Class 33

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Homework Check• Assignment:•  Chapter 7 – Exercise 7.68, 7.69 and 7.70• Reading:• Chapter 7 – p. 238-247

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Suggested Answer

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Do Now – 2Way Table

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Question: 1. P ( attended seminar and increased sales attended seminar)

2. P ( NOT attended seminar and increased sales NOT attended seminar)

Do Now- SimulationShooting Baskets

• Mr. Myer shot free throws at a 70% this season.

• How likely is it that he would make 7 shots in a row out of 10 shots? (Simulate 15 repetitions)

• Mr. Jameson shoots free throws at a 54% rate.

• How often would he make 7 in a row out of 10 shots? (Simulate 15 repetitions)

Use Tree Diagram

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Use Tree Diagram

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Use Tree Diagram and Random Number to do simulation

Question:

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 operation, 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 for at least five years are 70% for those with a new kidney and 50% for those who return to dialysis. Morris wants to know the probability that he will survive for at least 5 years.

Draw a Tree Diagram

Use Tree diagram to do simulation Step 1: Construct a tree diagram

0.9

0.6

0.1 0.5

0.4

0.7

0.5

0.3Survive

Survive

New Kidney

Survive

Die

Die

Dialysis

Die

Use Tree diagram to do simulation Step 2: Assign digits to outcome

Stage 1: 0 = die 1,2,3,4,5,6,7,8,9 = survive

Stage 2: 0, 1, 2, 3, 4, 5 = transplant succeeds 6, 7, 8, 9 = return to dialysisStage 3 with new kidney: 0, 1, 2, 3, 4, 5, 6 = survive for 5 years 7, 8, 9 = dieStage 3 with dialysis: 0, 1, 2, 3, 4 = survive for 5 years 5, 6, 7, 8, 9 = die

Use Tree diagram to do simulation Step 3: Use random number table to do simulations of several repetitions to determine the estimated probability of “survive five years”, each arranged verticallyFor example: random number: 17138 27584 252

Repetition1 Repetition 2 Repetition 3 Repetition 4

Stage 1 1 -> Survive 3 -> Survive 7 -> Survive 4 -> Survive

Stage 2 7 -> return to dialysis

8 -> return to dialysis

5 -> New Kidney 2 -> New Kidney

Stage 3 1 -> Survive 2 -> Survive 8 -> Die 5 -> Survive

Morris survives five years in 3 of the 4 repetitions.The estimated probability = 3/ 4

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7.7 Flawed Intuitive Judgments about Probability

Confusion of the Inverse

Example: Diagnostic Testing

Confuse the conditional probability “have the disease” given “a positive test result” -- P(Disease | Positive),with the conditional probability of “a positive test result” given “have the disease” -- P(Positive | Disease), also known as the sensitivity of the test.

Often forget to incorporate the base rate for a disease.

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Specific People versus Random Individuals

• In long run, about 50% of marriages end in divorce.

• At the beginning of a randomly selected marriage, the probability it will end in divorce is about 0.50.

Does this statement apply to you personally? If you have had a terrific marriage for 30 years, your probability of ending in divorce is surely less than 50%.

The chance that your marriage will end in divorce is 50%.

Two correct ways to express the aggregate divorce statistics:

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Coincidences

Example 7.33 Winning the Lottery Twice

A coincidence is a surprising concurrence of events, perceived as meaningfully related, with no apparent causal connection.

In 1986, Ms. Adams won the NJ lottery twice in a short time period. NYT claimed odds of one person winning the top prize twice were about 1 in 17 trillion. Then in 1988, Mr. Humphries won the PA lottery twice.

1 in 17 trillion = probability that a specific individual who plays the lottery exactly twice will win both times.

Millions of people play the lottery. It is not surprising that someone, somewhere, someday would win twice.

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The Gambler’s Fallacy

• Primarily applies to independent events.

• Independent chance events have no memory.

Example:

Making ten bad gambles in a row doesn’t change the probability that the next gamble will also be bad.

The gambler’s fallacy is the misperception of applying a long-run frequency in the short-run.

Homework• Assignment:•  Chapter 7 – Exercise 7.83, 7.85 and 7.89• Reading:• Chapter 7 – p. 238-247

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