Let's solve each part step by step. a) State the random variable. The random variable X X = the number of heads flipped when you flip a coin three times. b) Write the probability distribution for the number of heads. Let's list out all possible outcomes when flipping a coin 3 times: HHH HHT HTH HTT THH THT TTH TTT There are 2 3 = 8 2 3 =8 possible outcomes. Let's count the number of heads ( X X): 0 heads: TTT (1 way) 1 head: HTT, THT, TTH (3 ways) 2 heads: HHT, HTH, THH (3 ways) 3 heads: HHH (1 way) Now, calculate the probability for each value of X X: P ( X = 0 ) = 1 8 P(X=0)= 8 1 P ( X = 1 ) = 3 8 P(X=1)= 8 3 P ( X = 2 ) = 3 8 P(X=2)= 8 3 P ( X = 3 ) = 1 8 P(X=3)= 8 1 Fill out the table: X P(X) 0 1/8 1 3/8 2 3/8 3 1/8 c) If you were to draw a histogram for the number of heads, what shape would it be? The distribution is symmetric (centered at 1 and 2). d) Find the mean number of heads. The mean for a discrete probability distribution is: E [ X ] = X P ( X ) E[X]=XP(X) Plug in the values: E [ X ] = 0 1 8 + 1 3 8 + 2 3 8 + 3 1 8 = 0 + 3 8 + 6 8 + 3 8 = 3 + 6 + 3 8 = 12 8 = 1.5 E[X]=0 8 1 +1 8 3 +2 8 3 +3 8 1 =0+ 8 3 + 8 6 + 8 3 = 8 3+6+3 = 8 12 =1.5 Summary Table and Answers X P(X) 0 1/8 1 3/8 2 3/8 3 1/8 c) Histogram shape: Symmetric d) Mean number of heads: 1.5 find the standard deviaton