Question

five students, adriana, ben, chandra, diana, and ernesto, would each like one of the four spots at the regional science fair. their names are placed in a hat, and four names are chosen at random to decide who attends the fair. what is the theoretical probability that chandra will be chosen as one of the science fair participants? 60% 25% 80% 75%

Explanation:

Step1: Determine total and favorable cases

Total students: 5. We choose 4. The number of ways to choose 4 out of 5 is equivalent to the number of ways to leave out 1 (since choosing 4 to include is the same as choosing 1 to exclude). The number of ways Chandra is chosen is equal to the number of ways to choose the remaining 3 from the other 4 (since Chandra is included, we pick 3 more from 4). Alternatively, the probability that Chandra is chosen is 1 minus the probability she is not chosen. The probability she is not chosen: there's 1 way to not choose her (leave her out) and 4 ways to choose the others? Wait, no. The total number of possible groups of 4 is \( \binom{5}{4} = 5 \) (since choosing 4 from 5 is 5 combinations: each combination leaves out one student). The number of groups that include Chandra: if Chandra is included, we need to choose 3 more from the remaining 4, so \( \binom{4}{3} = 4 \). So the probability is \( \frac{4}{5} = 0.8 = 80\% \). Wait, no, wait. Wait, total number of ways to choose 4 names: since there are 5 students, and we pick 4, the number of possible outcomes is 5 (because each outcome is a group of 4, which is the same as excluding 1 student; so excluding Adriana, Ben, Chandra, Diana, or Ernesto: 5 possibilities). The number of favorable outcomes (Chandra is included) is the number of groups that include Chandra, which is the number of ways to exclude one of the other 4 students. So excluding Adriana, Ben, Diana, or Ernesto: 4 possibilities. So probability is \( \frac{4}{5} = 0.8 = 80\% \). Wait, but let's check another way. The probability that Chandra is chosen: when we pick 4 names, the chance that Chandra is in the 4 is equal to the number of ways to have Chandra plus 3 others, divided by total number of ways to pick 4. Total ways: \( C(5,4) = 5 \). Favorable ways: \( C(4,3) = 4 \) (since we must include Chandra, so choose 3 from the remaining 4). So \( 4/5 = 0.8 = 80\% \). So the correct answer is 80%.

Step2: Match with the options

The options include 60%, 25%, 80%, 75%. So 80% is the correct one.

Answer:

80%