Probability Question 6

Mating eagle pairs typically have two baby eagles (called eaglets). When there are two eaglets, the parents always feed the older eaglet until it has had its fill, and then they feed the younger eaglet. This results in an unequal chance of survival for the two eaglets. Suppose that the older eaglet has a 50 percent chance of survival. If the older eaglet survives, the younger eaglet has a 10 percent chance of survival. If the older eaglet does not survive, the younger eaglet has a 30 percent chance of survival. Let $X$ be the number of eaglets that survive. Which of the following tables shows the probability distribution of $X$ ?

\def\arraystretch{1.5} \begin{array}{|c||c|c|c|} \hline x & 0 & 1 & 2 \\ \hline p(x) & \frac{1}{3} & \frac{1}{3} &\frac{1}{3} \\ \hline \end{array}

\def\arraystretch{1.5} \begin{array}{|c||c|c|c|} \hline x & 0 & 1 & 2 \\ \hline p(x) & \frac{1}{4} & \frac{1}{2} &\frac{1}{4} \\ \hline \end{array}

\def\arraystretch{1} \begin{array}{|c||c|c|c|} \hline x & 0 & 1 & 2 \\ \hline p(x) & 0.35 & 0.60 & 0.05 \\ \hline \end{array}

\def\arraystretch{1} \begin{array}{|c||c|c|c|} \hline x & 0 & 1 & 2 \\ \hline p(x) & 0.05 & 0.90 & 0.05 \\ \hline \end{array}

Approach

The chance of 0 eaglets surviving can be found by taking the probability that the older eaglet does not survive and multiplying that by the probability that the younger eaglet does not survive.

P(\text{0 Survive}) = (0.50)(0.70) = 0.35

We can stop here since there is only one answer choice with this option. If we wanted to know the chances that both eagles survive:

P(\text{2 Survive}) = (0.50)(0.10) = 0.05

There are two cases where one eaglet can remain:

P(\text{Only older survives}) = (0.50)(0.90) = 0.45

P(\text{Only younger survives}) = (0.50)(0.30) = 0.15

P(\text{One survives}) = 0.45+0.15 = 0.60