Question

3. A lady claimed that she is able to tell whether the tea was added first or the milk was added first to a cup. To test this idea, a statistician proposed to give her ten cups of tea, each made in random order (tea first or milk first) without telling her which is which. Assume each cup is independent. Let X equal the number of cups that the lady identified correctly. (a) Suppose that she is just randomly guessing, with a 50-50 percent chance. Find P(X = 7). (b) Suppose again that she is randomly guessing, with a 50-50 percent chance. This experiment will be stopped early if she cannot correctly identify at least one cup among the first three cups. What is the probability that this experiment continues beyond three cups? (c) Suppose she is indeed able to tell 9% of the time. Find the probability she correctly identifies at least 7 cups.

          3. A lady claimed that she is able to tell whether the tea was added first or the milk was added first to a cup. To test this idea, a statistician proposed to give her ten cups of tea, each made in random order (tea first or milk first) without telling her which is which. Assume each cup is independent. Let X equal the number of cups that the lady identified correctly. (a) Suppose that she is just randomly guessing, with a 50-50 percent chance. Find P(X = 7). (b) Suppose again that she is randomly guessing, with a 50-50 percent chance. This experiment will be stopped early if she cannot correctly identify at least one cup among the first three cups. What is the probability that this experiment continues beyond three cups? (c) Suppose she is indeed able to tell 9% of the time. Find the probability she correctly identifies at least 7 cups.

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3. A lady claimed that she is able to tell whether the tea was added first or the milk was added first to a cup. To test this idea, a statistician proposed to give her ten cups of tea, each made in random order (tea first or milk first) without telling her which is which. Assume each cup is independent. Let X equal the number of cups that the lady identified correctly. (a) Suppose that she is just randomly guessing, with a 50-50 percent chance. Find P(X = 7). (b) Suppose again that she is randomly guessing, with a 50-50 percent chance. This experiment will be stopped early if she cannot correctly identify at least one cup among the first three cups. What is the probability that this experiment continues beyond three cups? (c) Suppose she is indeed able to tell 9% of the time. Find the probability she correctly identifies at least 7 cups.

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Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Jacob Fry

10/26/2022

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Step 1: Calculate the probability of correctly identifying 7 cups out of 10 when randomly guessing with a 50-50 chance. Show more…

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A lady claimed that she is able to tell whether the tea was added first or the milk was added first to a cup. To test this idea, a statistician proposed to give her ten cups of tea, each made in random order (tea first or milk first), without telling her which is which. Assume each cup is independent. Let X equal the number of cups that the lady identified correctly. (a) Suppose that she is just randomly guessing with a 50-50 percent chance. Find P(X = 7). (b) Suppose again that she is randomly guessing, with a 50-50 percent chance. This experiment will be stopped early if she cannot correctly identify at least one cup among the first three cups. What is the probability that this experiment continues beyond three cups? (c) Suppose she is indeed able to tell 9% of the time. Find the probability she correctly identifies at least 7 cups.

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Transcript

00:01 So in this question we have somebody who claims to be able to tell whether tea is added first or milk is added first to a cup.

00:10 And so let's assume.

00:12 So x is going to be the number of cups correctly identified.

00:28 Then first of all, if we assume that she's randomly guessing, then x is going to be binomially distributed.

00:37 And she's given 10 cups of tea.

00:39 So there's 10 trials.

00:41 And if she's randomly guessing, she has a half chance of success.

00:46 So that means that the probability, the pmf of the binomial distribution, we know, is 10 choose x.

00:56 And when we have a half as the probability of success, we just raise a half to the power of 10.

01:03 So the probability that x is equal to 7 is going to be 10 choose 7 times 1⁄2 to the power of 10.

01:13 So 1 over 2 to the power of 10 times 10 choose 7 is 15 over 128, which is approximately 0 .117.

01:33 Okay, so now again, she's randomly guessing, and the experiment will be stopped early if she cannot correctly identify at least one cup among the first three cups.

01:44 So what's the probability that we continue after three cups? well, she needs to identify at least one of the first three.

02:01 So let y be the number of the first three identified correctly.

02:13 Then y is binomially distributed with n equals three and p equals a half.

02:19 So the probability that y is greater than or equal to 1 is the sum from y equals 1 to 3 of 3 choose y times a half to the power of 3.

02:37 But what we also know is that the only chance that y cannot be greater than or equal to 1 is if y equals 0.

02:45 So this is 1 minus the probability that y is equal to 0.

02:48 So this is 1 minus 3 -2 -0 times 1 -half to the power of 3...

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