Question

initial knowledge check question 21 an ordinary (fair) coin is tossed 3 times. outcomes are thus triples of \heads\ (h) and \tails\ (t) which we write hth, ttt, etc. for each outcome, let n be the random variable counting the number of heads in each outcome. for example, if the outcome is thh, then n(thh) = 2. suppose that the random variable x is defined in terms of n as follows: x = 2n - 4. the values of x are given in the table below. outcome hth tht ttt hhh hht thh sth htt value of x 0 -2 -4 2 0 0 -2 -2 calculate the probabilities p(x = x) of the probability distribution of x. first, fill in the first row with the values of x. then fill in the appropriate probabilities in the second row. value x of x \boxed{} \boxed{} \boxed{} \boxed{} p(x = x) \boxed{} \boxed{} \boxed{} \boxed{}

Explanation:

Step1: List unique X values

From the table, unique $X$ values: $-4, -2, 0, 2$

Step2: Count total outcomes

Total coin toss outcomes: $2^3=8$

Step3: Count each X's frequency

• $X=-4$: 1 outcome ($ttt$)
• $X=-2$: 3 outcomes ($tht, tth, htt$)
• $X=0$: 3 outcomes ($hth, hht, thh$)
• $X=2$: 1 outcome ($hhh$)

Step4: Calculate each probability

$P(X=x)=\frac{\text{Frequency of }x}{\text{Total outcomes}}$

• $P(X=-4)=\frac{1}{8}$
• $P(X=-2)=\frac{3}{8}$
• $P(X=0)=\frac{3}{8}$
• $P(X=2)=\frac{1}{8}$

Answer:

Value $x$ of $X$	$-4$	$-2$	$0$	$2$