Discussion 2 (Purple Bẩby in Statopia) Let $X=\#$ of PURPLE in 6 births. Assume there is $40 \%$ chance of getting PURPLE in each birth in Statopia. 1. What is the probability distribution for $\mathrm{X}$, \# of Purple baby in 6 births? 2. What are the expected value (mean) and standard deviation for the number of PURPLE in 6 births? 3. What is the Probability that a family of 6 kids are all PURPLE?

Step 1 ：We are given a problem where the probability of a baby being born PURPLE in Statopia is 40%. We are asked to find the probability distribution of the number of PURPLE babies in 6 births. This is a binomial distribution problem, where the number of trials is 6 (the number of births), the probability of success (a PURPLE baby) is 0.4, and we want to find the probability of each possible number of successes (from 0 to 6).

Step 2 ：The probability distribution for the number of PURPLE babies in 6 births is as follows: The probability of having 0 PURPLE babies is approximately 0.0467, the probability of having 1 PURPLE baby is approximately 0.1866, the probability of having 2 PURPLE babies is approximately 0.3110, the probability of having 3 PURPLE babies is approximately 0.2765, the probability of having 4 PURPLE babies is approximately 0.1382, the probability of having 5 PURPLE babies is approximately 0.0369, and the probability of having 6 PURPLE babies is approximately 0.0041.

Step 3 ：Final Answer: The probability distribution for the number of PURPLE babies in 6 births is approximately \(\boxed{[0.0467, 0.1866, 0.3110, 0.2765, 0.1382, 0.0369, 0.0041]}\).