Question

According to a government commission, 70% of all the nation's households have vcrs. in a random sample of 15 households, what is the probability that have at least 13 vcrs? round to four decimal places.

Answer 1

To find the probability of at least 13 out of 15 households having VCRs when 70% of households do, use the binomial distribution formula by subtracting the cumulative probability of having 0 to 12 households from 1.

Step-by-step explanation:

The question pertains to the concept of a binomial distribution, which is a part of probability theory in mathematics. To calculate the probability that at least 13 out of 15 randomly selected households have VCRs, given that the probability of a household having a VCR is 70%, we use the binomial probability formula:

P(X ≥ k) = 1 - ∑i=0k-1 [C(n, i) × pi × (1-p)n-i]

Where:

• n is the number of trials (households), which in this case is 15.
• X is the random variable representing the number of households with VCRs.
• k is the number of successful outcomes, here being at least 13, so we calculate for k=13 and k=14.
• p is the probability of success on a single trial, which is 0.70.
• C(n, i) is the combination of n taken i at a time.

Using the formula, we sum up the probabilities of having exactly 0 to 12 households with VCRs and subtract this sum from 1 to find the probability of having at least 13 households with VCRs.

Answer 2

Number of trials, n = 15
Expected success, r = 13
Probability of having a VCR, p = 70% = 0.7
Probability of not having a VCR, q = 1 - p = 0.3

According to the binomial distribution, the probability of 13 out of 15 households having a VCR is
₁₅C₁₃ p¹³ q¹⁵⁻¹³
= [15!/(13!2!)]*(0.7¹³)*(0.3²)
= 0.0916

Answer: 0.0916 (or 9.16%)