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You are taking a multiple choice test for which you have mastered 70% of the material. Assume this means that you have a 0.7 chance of knowing the answer to a random test question, and that if you don’t know the answer to a question then you randomly select among the four answer choices. Finally, assume that this holds for each question, independent of the others. (a) What is your expected score (as a percent) on the exam? Let p be the probability of getting a random question correct. You likely found p in part (a), but in any case you should assume 0.7 < p < 0.9. In parts (b), (c) and (d) you can just use the letter p for this probability. (b) If the test has 10 questions, what is the probability you score 90% or higher? (Do not simplify the expression you get.) (c) What is the probability you get the first 6 questions on the exam correct? (Again, do not simplify the expression you get.) (d) Suppose you need a 90% to keep your scholarship. Would you rather have a test with 10 or 100 questions? Why? (Your answer only needs to be one sentence long.)

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You are taking a multiple choice test for which you have mastered 70% of the material.

Assume this means that you have a 0.7 chance of knowing the answer to a random test

question, and that if you don’t know the answer to a question then you randomly select

among the four answer choices. Finally, assume that this holds for each question, independent of the others.

(a) What is your expected score (as a percent) on the exam?
Let p be the probability of getting a random question correct. You likely found p in part
(a), but in any case you should assume 0.7 < p < 0.9.
In parts (b), (c) and (d) you can just use the letter p for this probability.
(b) If the test has 10 questions, what is the probability you score 90% or higher? (Do not
simplify the expression you get.)
(c) What is the probability you get the first 6 questions on the exam correct? (Again, do
not simplify the expression you get.)
(d) Suppose you need a 90% to keep your scholarship. Would you rather have a test with
10 or 100 questions? Why? (Your answer only needs to be one sentence long.)

I have a bag with 3 coins in it. One of them is a fair coin, but the others are biased trick coins. When flipped, the three coins come up heads with probability 0.5, 0.6, 0.1 respectively. Suppose that I pick one of these three coins uniformly at random and flip it three times. (a) What is P(HT T)? (That is, it comes up heads on the first flip and tails on the second and third flips.) (b) Assumimg that the three flips, in order, are HT T, what is the probability that the coin that I picked was the fair coin? (Remember, there is no need to simplify fractions.)

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I have a bag with 3 coins in it. One of them is a fair coin, but the others are biased

trick coins. When flipped, the three coins come up heads with probability 0.5, 0.6, 0.1

respectively.

Suppose that I pick one of these three coins uniformly at random and flip it three times.

(a) What is P(HT T)? (That is, it comes up heads on the first flip and tails on the second
and third flips.)
(b) Assumimg that the three flips, in order, are HT T, what is the probability that the coin
that I picked was the fair coin?
(Remember, there is no need to simplify fractions.)

Let X1, X2, . . . , X81 be i.i.d., each with expected value µ = E(Xi) = 5, and variance σ2 = Var(Xi) = 4. Approximate P(X1 + X2 +· · · X81 > 369), using the central limit theorem.

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Let X1, X2, . . . , X81 be i.i.d., each with expected value
µ = E(Xi) = 5, and variance σ2 = Var(Xi) = 4. Approximate P(X1 + X2 +· · · X81 > 369),
using the central limit theorem.

Suppose that X ∼ Bin(n, 0.5). Find the probability mass function of Y = 2X.

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Suppose that X ∼ Bin(n, 0.5). Find the probability mass function of Y = 2X.

Suppose that P(A) = 0.4, P(B) = 0.3 and P((A ∪ B)C) = 0.42. Are A and B independent?

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Suppose that P(A) = 0.4, P(B) = 0.3 and P((A ∪ B)C) = 0.42. Are A and B
independent?

A multiple choice exam has 4 choices for each question. A student has studied enough so that the probability they will know the answer to a question is 0.5, the probability that they will be able to eliminate one choice is 0.25, otherwise all 4 choices seem equally plausible. If they know the answer they will get the question right. If not they have to guess from the 3 or 4 choices. As the teacher you want the test to measure what the student knows. If the student answers a question correctly what’s the probability they knew the answer?

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A multiple choice exam has 4 choices for each question. A student has studied enough so
that the probability they will know the answer to a question is 0.5, the probability that they
will be able to eliminate one choice is 0.25, otherwise all 4 choices seem equally plausible.
If they know the answer they will get the question right. If not they have to guess from the
3 or 4 choices.
As the teacher you want the test to measure what the student knows. If the student answers
a question correctly what’s the probability they knew the answer?

Suppose you are taking a multiple-choice test with c choices for each question. In answering a question on this test, the probability that you know the answer is p. If you don’t know the answer, you choose one at random. What is the probability that you knew the answer to a question, given that you answered it correctly?

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Suppose you are taking a multiple-choice test with c choices for each question. In
answering a question on this test, the probability that you know the answer is p. If you
don’t know the answer, you choose one at random. What is the probability that you knew
the answer to a question, given that you answered it correctly?

Let A and B be two events. Suppose the probability that neither A or B occurs is 2/3. What is the probability that one or both occur?

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Let A and B be two events. Suppose the probability that neither A or B occurs is 2/3.
What is the probability that one or both occur?

The grades of a group of 1000 students in an exam are normally distributed with a mean of 70 and a standard deviation of 10. A student from this group is selected randomly. a) Find the probability that his/her grade is greater than 80. b) Find the probability that his/her grade is less than 50. c) Find the probability that his/her grade is between 50 and 80. d) Approximately, how many students have grades greater than 80?

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The grades of a group of 1000 students in an exam are normally distributed with a mean of 70 and a standard deviation of 10. A student from this group is selected randomly.
a) Find the probability that his/her grade is greater than 80.
b) Find the probability that his/her grade is less than 50.
c) Find the probability that his/her grade is between 50 and 80.
d) Approximately, how many students have grades greater than 80?

In a school, 60% of pupils have access to the internet at home. A group of 8 students is chosen at random. Find the probability that a) exactly 5 have access to the internet. b) at least 6 students have access to the internet.

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In a school, 60% of pupils have access to the internet at home. A group of 8 students is chosen at random. Find the probability that
a) exactly 5 have access to the internet.
b) at least 6 students have access to the internet.