Question

Consider tossing a fair6-sideddie with the sample space S=1,2,3,4,5,6

(a)What are the elementary events? What are the elementary probabilities? Does S consist of equally-likely outcomes?

(b)What is the event E that the outcome of the experiment is even? What is the event F that the outcome is odd? Are E and F independent? What is the conditional probability P(E|F)?

(c) Consider the variable X:S→R of the probability space, where R is the set of real numbers, and X(k)=k for k=1, 2, 3, 4, 5, 6. Calculate the expectation, variance, and standard deviation of X. Show the steps of your calculations.

(d) Now consider tossing two fair 6-sided dice together and record the product of the two numbers obtained. What is the sample space of this experiment? Does the sample space consist of equally-likely outcomes? Explain why.

(e) Note that all the numbers in this problem are made-up.

Suppose the probability of a random person being infected by COVID-19 is 0.001 (or 0.1%). The CDC is developing a new test and in a recent study it was report that:

1. The new test is 90% accurate when applied to a subject that is known to have COVID-19.
2. The new test is 95% accurate when applied to a subject that is known to be healthy.

What is the probability that a subject testing positive has COVID-19?

Answer 1

(a)

The elementary events are the die coming up 1,2,3,4,5 or 6. As each are equally likely because the die is unbiased, and there are six possibilities, each one has probability 1/6. S does consist of equally likely outcomes.

(b)

There are 3 choices of even (2,4, and 6), out of 6, hence the probability is 1/2. For odd as well, there are three choices, hence probability is 1/2.

E and F are not independent, as if F happens i.e. the die comes up odd, it precludes E happening as it cannot be even. , as if F happens, E cannot happen.

(c)

To calculate the expectation
, as 1 to 6 are the only values with non-zero probability, and each of them have probability 1/6.

Thus .

For variance, , as 1 to 6 are the only values. Use the formula for sum of squares of natural numbers to get
.

.

(d)

The possible products of outcomes of two die are
.

Hence the sample space is .

These are not equally likely as has two combinations from which it can happen, while has only one combination, hence 12 will have more probability than 3.

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