Question

Senior Citizens In the 2010 US Census, we learn that \(13 \%\) of all people in the US are 65 years old or older. If we take a random sample of 10 people, what is the probability that 3 of them are 65 or older? That 4 of them are 65 or older?

Short answer

After evaluating the binomial probabilities you will find the probability that 3 out of 10 people are 65 or older and the probability that 4 out of 10 people are 65 or older. These probabilities have to be calculated separately.

Step-by-step solution

Calculate combinations

We start off by calculating the number of combinations of 10 people that include 3 and 4 individuals respectively. We use the combination formula \(C(n, k) = n! / [k!(n-k)!]\) where \(n\) is the total number of people and \(k\) is the number of successful outcomes. Our two combinations therefore are \(C(10, 3)\) and \(C(10, 4)\).

Calculate probabilities

We then use the binomial probability formula \(P(k; n, p) = C(n, k) * (p^k) * ((1-p)^(n-k))\). Substituting the values into the formula we have \(P(3; 10, .13) = C(10, 3) * (0.13^3) * ((1-0.13)^(10-3))\) for the first probability and \(P(4; 10, .13) = C(10, 4) * (0.13^4) * ((1-0.13)^(10-4))\) for the second probability.

Evaluate probabilities

Finally we can evaluate to get the probabilities. For our first probability we solve \(P(3; 10, .13)\) to get the probability that 3 out of 10 people are 65 or older. Then we solve \(P(4; 10, .13)\) to get the probability that 4 out of 10 people are 65 or older.