The Denver Post stated that 80% of all new products introduced in grocery stores fail (are taken off the market) within 2 years. If a grocery chain introduces 66 new products, what is the probability that within 2 years , 7 or more fail? What is the probability that within 2 years , 58 or fewer fail?
What is the probability that 15 or more succeed?
What is the probability that fewer than 10 succeed?
To answer these questions, we can use the binomial probability formula. The binomial probability formula calculates the probability of a certain number of successes (or failures) in a fixed number of independent trials, with each trial having the same probability of success.
The binomial probability formula is: P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
- P(x) is the probability of getting exactly x successes
- n is the number of trials or observations (in this case, the number of new products introduced)
- x is the number of desired successes (or failures)
- p is the probability of success in a single trial (in this case, the probability of a new product failing)
To answer the specific questions:
1. Probability that 7 or more fail:
To calculate this, we need to calculate the probabilities of 7, 8, 9, ..., and 66 failing, and then sum them up.
P(7 or more fail) = P(7 fails) + P(8 fails) + P(9 fails) + ... + P(66 fails)
2. Probability that 58 or fewer fail:
Similarly, we need to calculate the probabilities of 0, 1, 2, ..., and 58 failing, and then sum them up.
P(58 or fewer fail) = P(0 fails) + P(1 fails) + P(2 fails) + ... + P(58 fails)
3. Probability that 15 or more succeed:
This is the complement of the probability that 14 or fewer succeed, which can be calculated using the same approach as above.
P(15 or more succeed) = 1 - (P(0 succeeds) + P(1 succeeds) + P(2 succeeds) + ... + P(14 succeeds))
4. Probability that fewer than 10 succeed:
This is the complement of the probability that 9 or more succeed, which can be calculated similarly.
P(fewer than 10 succeed) = 1 - (P(9 succeeds) + P(10 succeeds) + P(11 succeeds) + ... + P(66 succeeds))
To calculate each of these probabilities, we can use the binomial probability formula and substitute the appropriate values for n, x, and p. The value of n is 66 (the number of new products introduced), and p is 0.8 (the probability of failure).