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Chapter 2: Problem 3

In Exercise 2 , if the coin is assumed fair, then, for \(n=2\), what are the probabilities associated with the values that \(X\) can take on?

Short Answer

Expert verified
For a fair coin with \(n=2\), the probabilities associated with the values that \(X\) can take on are: P(X=0) = 1/4, P(X=1) = 1/2, and P(X=2) = 1/4.

Step by step solution

01

Identify possible outcomes and corresponding probabilities for a single flip

Since the coins is fair, the only two outcomes from a single coin flip are: - A head (H) with a probability of P(H) = 1/2 - A tail (T) with a probability of P(T) = 1/2
02

Determine possible outcomes for flipping the coin twice

Considering all the combinations from two flips, we have: - HH: P(HH) = P(H) 脳 P(H) = (1/2) 脳 (1/2) = 1/4 - HT: P(HT) = P(H) 脳 P(T) = (1/2) 脳 (1/2) = 1/4 - TH: P(TH) = P(T) 脳 P(H) = (1/2) 脳 (1/2) = 1/4 - TT: P(TT) = P(T) 脳 P(T) = (1/2) 脳 (1/2) = 1/4
03

Calculate the probability for each possible value of \(X\)

Now, we calculate probabilities for each possible value of \(X\): - X = 0: 0 heads; only outcome is TT. P(X=0) = P(TT) = 1/4 - X = 1: 1 head; outcomes are HT and TH. P(X=1) = P(HT) + P(TH) = 1/4 + 1/4 = 1/2 - X = 2: 2 heads; only outcome is HH. P(X=2) = P(HH) = 1/4
04

Summarize the probabilities for each possible value of \(X\)

The probabilities associated with the values that \(X\) can take on when flipping a fair coin twice are: - P(X=0) = 1/4 - P(X=1) = 1/2 - P(X=2) = 1/4

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Most popular questions from this chapter

A coin, having probability \(p\) of landing heads, is flipped until head appears for the \(r\) th time. Let \(N\) denote the number of flips required. Calculate \(E[N]\). Hint: There is an easy way of doing this. It involves writing \(N\) as the sum of \(r\) geometric random variables.

Let \(X\) represent the difference between the number of heads and the number of tails obtained when a coin is tossed \(n\) times. What are the possible values of \(X ?\)

Suppose a die is rolled twice. What are the possible values that the following random variables can take on? (i) The maximum value to appear in the two rolls. (ii) The minimum value to appear in the two rolls. (iii) The sum of the two rolls. (iv) The value of the first roll minus the value of the second roll.

An urn contains \(n+m\) balls, of which \(n\) are red and \(m\) are black. They are withdrawn from the urn, one at a time and without replacement. Let \(X\) be the number of red balls removed before the first black ball is chosen. We are interested in determining \(E[X] .\) To obtain this quantity, number the red balls from 1 to \(n\). Now define the random variables \(X_{i}, i=1, \ldots, n\), by $$ X_{i}=\left\\{\begin{array}{ll} 1, & \text { if red ball } i \text { is taken before any black ball is chosen } \\ 0, & \text { otherwise } \end{array}\right. $$ (a) Express \(X\) in terms of the \(X_{i}\). (b) Find \(E[X]\).

An urn contains \(2 n\) balls, of which \(r\) are red. The balls are randomly removed in \(n\) successive pairs. Let \(X\) denote the number of pairs in which both balls are red. (a) Find \(E[X]\). (b) Find \(\operatorname{Var}(X)\).
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