Question

You are bidding on six items available on an online shopping site. You think that for each bid you have a 25​% chance of winning​ it, and the outcomes of the six bids are independent events. Let X denote the number of winning bids out of the six items you bid on. Find the probabilities.

a. Explain why the distribution of X can be modeled by the binomial distribution.

b. Find the probability that you win exactly 4 bids.

c. Find the probability that you win 4 bids or fewer.

d. Find the probability that you win more than 4 bids.

Answer 1

a. The distribution of X can be modelled by the binomial distribution because the sample size is fixed, each observation is independent of the other, and each observation represents either success or failure. b. The calculation of probability is as shown below: p( X = 4) = C, x0.25* (1-0.25) 4 4!(6-4)! -- *(1-0.25) = 0.0330 Therefore, the probability that I win 4 bids is 0.0330.

c. The calculation of probability is as shown below: Cox0.250 x(1-0.25). + *C+0.254x(1-0.25)++ p(X + 4) = C, 0.252x(1-0.25) $C, x0.25* *(1-0.25)- + Cx0.25*x(1-0.25)64 + 0!(6-0)2X0.25° x(1–0,25) + 1!(6-1)! *0.25 1, x0.25*x(1–0,25)(4+ 2462), *0,25°x(1–0.25)+ 3!(6-3)*0.25*<(1–0.25)- + 4!(6-4)7X0.25* x(1–0.25)84 6! =0.9954 Therefore, the probability that I win 4 bids or fewer is 0.9954. d. The calculation of probability is as shown below: p(X > 4)=1-p(X 34) =1-0.995 = 0.0046 Therefore, the probability that I win more than 4 bids is 0.0046.

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