Question

39.7​% of consumers believe that cash will be obsolete in the next 20 years. Assume that 7 consumers are randomly selected. Find the probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years.

Answer 1

The probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years is approximately 0.374 or 37.4%.

Using the binomial probability distribution

The probability for each possible number of successes (x) is

For x = 0 (no consumers believing cash will be obsolete):

P(x = 0) =
`q^n * nCx` =
`0.603^7 * 7C0` ≈ 0.029

For x = 1 (one consumer believing cash will be obsolete)

P(x = 1) =
`q^((n-x)) * p^x * nCx = 0.603^6 * 0.397^1 * 7C1`≈ 0.126

For x = 2 (two consumers believing cash will be obsolete)

P(x = 2) =
`q^((n-x)) * p^x * nCx = 0.603^5 * 0.397^2 * 7C2` ≈ 0.219

Sum the probabilities for "fewer than 3" successes

P(x < 3) = P(x = 0) + P(x = 1) + P(x = 2) ≈ 0.029 + 0.126 + 0.219 ≈ 0.374

Therefore, the probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years is approximately 0.374 or 37.4%.