Question

Ten students begin college at the same time. The probability of graduating in four years is 63%. Which expanded expression shows the first and last terms of the expression used to find the probability that at least six will graduate in four years?

a. ( 10.6 ) × ( 0.63 ) 6 × ( 0.37 ) 4 . . . . ( 10.10 ) × ( 0.63 ) 10 × ( 0.37 ) 0 = 0.706
b. ( 10.5 ) × ( 0.63 ) 5 × ( 0.37 ) 5 . . . . ( 10.10 ) × ( 0.63 ) 10 × ( 0.37 ) 0 = 0.88
c. ( 10.0 ) × ( 0.63 ) 0 × ( 0.37 ) 10 . . . . ( 10.6 ) × ( 0.63 ) 6 × ( 0.37 ) 4 = 0.540

Answer 1

In a binomial distribution problem, the probability of at least six out of ten students graduating in four years, when the probability of an individual graduating is 63%, is represented by the sum of the probabilities of exactly six to ten students graduating. The correct expanded expression has the binomial terms (10 choose 6) times 0.63 to the power of 6 times 0.37 to the power of 4 for the first term and (10 choose 10) times 0.63 to the power of 10 times 0.37 to the power of 0 for the last term. The provided options are incorrect but option c is closest in structure.

Step-by-step explanation:

The probability of ten students graduating in four years with an individual probability of 63% is a binomial distribution problem. To calculate the probability that at least six will graduate, you would sum up the probabilities of exactly six, seven, eight, nine, and ten students graduating. Since the question asks only for the first and the last terms of this expression:

• The first term represents the probability of exactly six students graduating, which uses the binomial coefficient (10 choose 6), the probability of graduation 0.63 to the power of 6, and the probability of not graduating 0.37 to the power of 4.
• The last term represents the probability of all ten students graduating, which uses the binomial coefficient (10 choose 10), the probability of graduation 0.63 to the power of 10, and the probability of not graduating 0.37 to the power of 0.

Therefore, the expanded expression showing only the first and last terms should be:

(10 choose 6) × (0.63)6 × (0.37)4 .... (10 choose 10) × (0.63)10 × (0.37)0

Looking at the provided options, none of the expressions correspond exactly to the correct form for the first and last terms as explained above. However, option c resembles the correct form with the right coefficients and powers of probabilities for the first and last terms, but the terms are provided in reverse order, hence we know the correct answer structure but the options provided appear to contain errors.