Question

The probability that the first record of a singing group will be a hit is 0.24. If their first record is a hit, so are all their subsequent records. If their first record is not a hit, the probability of their second and all subsequent ones being hits is 0.12. If the first two records are not hits, the probability that the third is a hit is 0.06. Find the probability that a group has at least 1 hit in their first three records.

P(at least 1 hit in first three records) =
(Simplify your answer. Type an integer or a decimal. Round to 6 places.)

Answer 1

To calculate the probability that a group has at least one hit in their first three records, we consider the complementary probability of having no hits and subtract it from 1. After calculating the complementary probability, the final probability of at least one hit is approximately 0.994528.

Step-by-step explanation:

The probability that the first record of a singing group will be a hit is 0.24.

To find the probability that the group has at least 1 hit in their first three records, we need to consider the complementary scenario where the group has no hits in the first three records and subtract this from 1.

Let H denote a hit and N denote not a hit. The scenarios where they have no hits are:
First record is not a hit: P(N) = 1 - 0.24

= 0.76

Second record is not a hit given the first record wasn't a hit: P(N|first N) = 0.12

Third record is not a hit given the first two records weren't hits: P(N|first two N) = 0.06
The probability of no hits in the first three records is the product of these probabilities: P(NNN) = 0.76 × 0.12 × 0.06.

So, the probability that at least one record is a hit is: P(at least one hit) = 1 - P(NNN).
Calculating P(NNN) = 0.76 × 0.12 × 0.06

= 0.005472.
Therefore, P(at least one hit) = 1 - 0.005472

= 0.994528.

Rounding to six places, the answer is 0.994528.