the sample space, s, of a coin being tossed three times is shown below, where h and t denote the coin…

the sample space, s, of a coin being tossed three times is shown below, where h and t denote the coin landing on heads and tails respectively. s = {hhh, hht, hth, htt, thh, tht, tth, ttt} let x = the number of times the coin comes up heads. what is the probability distribution for the number of heads occurring in three coin tosses?

the sample space, s, of a coin being tossed three times is shown below, where h and t denote the coin landing on heads and tails respectively. s = {hhh, hht, hth, htt, thh, tht, tth, ttt} let x = the number of times the coin comes up heads. what is the probability distribution for the number of heads occurring in three coin tosses?

Answer

Explanation:

Step1: Count total outcomes

The sample - space $S={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}$ has $n(S) = 8$ total outcomes.

Step2: Calculate $P(X = 0)$

The event $X = 0$ (no heads) corresponds to the outcome $TTT$. So $n(X = 0)=1$, and $P(X = 0)=\frac{n(X = 0)}{n(S)}=\frac{1}{8}$.

Step3: Calculate $P(X = 1)$

The events with 1 head are $HTT$, $THT$, $TTH$. So $n(X = 1)=3$, and $P(X = 1)=\frac{n(X = 1)}{n(S)}=\frac{3}{8}$.

Step4: Calculate $P(X = 2)$

The events with 2 heads are $HHT$, $HTH$, $THH$. So $n(X = 2)=3$, and $P(X = 2)=\frac{n(X = 2)}{n(S)}=\frac{3}{8}$.

Step5: Calculate $P(X = 3)$

The event $X = 3$ (3 heads) corresponds to the outcome $HHH$. So $n(X = 3)=1$, and $P(X = 3)=\frac{n(X = 3)}{n(S)}=\frac{1}{8}$.

Answer:

$X$ 0 1 2 3
$P(X)$ $\frac{1}{8}$ $\frac{3}{8}$ $\frac{3}{8}$ $\frac{1}{8}$