Question

Back when gyms were open, I purchased an unlimited 4 month cycling membership, and I’d love to say I used it daily, but in reality I only attend 62% of the time. If there are 120 days in my 4 months, what is the probability I used my pass …

A) Exactly 75 times.

B) Under 60 times.        

C) At least 80 times.                 

D) Between 65 and 85 times, inclusively.

Answer 1

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This is an application of binomial distribution , with parameters p = .62, n = 120 , B(n,p) = B(120, 0.62)

a. Exactly 75 times

= P(x,n,p)
= P(75, 120, .62)
= nCx*p^x *(1-p)^(n-x)
= 120C75*.62^75 * .38^45
= 0.0746

b. P(X<60)
= P(X<=59)
= BINOM.DIST(59,120,.62,TRUE) [Excel formula or you can use the calculator function binomcdf]
= 0.0028

c. P(X>=80)
= 1-P(X<80)
= 1-P(X<=79)
= 1- 0.0524
= 0.9476

d. P(65<=X<=85)
= P(X<=85) - P(X<=64)
= BINOM.DIST(85,120,.62,TRUE) - BINOM.DIST(64,120,.62,TRUE))
= 0.9830 - 0.0324
= 0.9506