A certain type of new business succeeds 70% of the time. Suppose that four such businesses open (where they do not compete with each other, so it is reasonable to believe that their relative successes would be independent). The probability that at least two businesses succeeds is?
To calculate the probability that at least two out of four businesses succeed, we can calculate the probability that exactly two, exactly three, and exactly four businesses succeed and add them together.
Let's calculate each scenario:
1. Probability that exactly two businesses succeed:
To calculate this, we need to choose 2 businesses out of 4 to succeed, and the probability that these two businesses succeed is 0.7^2 * 0.3^2 = 0.441.
Using the binomial coefficient formula, the calculation is:
4 choose 2 = 4! / (2!(4-2)!) = 6
Therefore, the probability that exactly two businesses succeed is 6 * 0.441 = 2.646.
2. Probability that exactly three businesses succeed:
To calculate this, we need to choose 3 businesses out of 4 to succeed, and the probability that these three businesses succeed is 0.7^3 * 0.3 = 0.441.
Using the binomial coefficient formula, the calculation is:
4 choose 3 = 4! / (3!(4-3)!) = 4
Therefore, the probability that exactly three businesses succeed is 4 * 0.441 = 1.764.
3. Probability that all four businesses succeed:
The probability that all four businesses succeed is 0.7^4 = 0.2401.
Since these three scenarios are mutually exclusive, we can add their probabilities to find the total probability that at least two businesses succeed:
2.646 + 1.764 + 0.2401 = 4.6501
Therefore, the probability that at least two businesses succeed out of the four is 4.6501 or approximately 46.5%.