Question

a coin is tossed three times, with possible outcomes: {hhh, hht, hth, thh, htt, tht, tth, ttt}
identify the graph of the probability distribution for the random variable representing the number of heads.

Explanation:

Step1: Determine possible values of X

When a coin is tossed 3 times, the number of heads (X) can be 0, 1, 2, or 3. So we can eliminate the graph with X values 1,2,3,4 (the second graph) since 4 heads is impossible in 3 tosses.

Step2: Calculate probabilities for each X

• For \( X = 0 \) (TTT): There is 1 outcome. Probability \( P(0)=\frac{1}{8} = 0.125 \)
• For \( X = 1 \) (HTT, THT, TTH): 3 outcomes. Probability \( P(1)=\frac{3}{8}=0.375 \)
• For \( X = 2 \) (HHT, HTH, THH): 3 outcomes. Probability \( P(2)=\frac{3}{8}=0.375 \)
• For \( X = 3 \) (HHH): 1 outcome. Probability \( P(3)=\frac{1}{8}=0.125 \)

Step3: Analyze the remaining graphs

• The first graph has equal heights for 0,1,2,3 which would imply equal probabilities, but our calculated probabilities have \( P(0)=P(3)=0.125 \) and \( P(1)=P(2)=0.375 \), so it's incorrect.
• The third graph (bottom - left) has heights for 0 and 3 around 0.125 (matches \( \frac{1}{8} \)) and heights for 1 and 2 around 0.375 (matches \( \frac{3}{8} \)), which matches our calculated probabilities.

Answer:

The graph with X - axis values 0,1,2,3 (the bottom - left graph) where \( P(X = 0)=P(X = 3)\approx0.125 \) and \( P(X = 1)=P(X = 2)\approx0.375 \)