A consultant knows that it will cost him $10,000 to fulfill a particular contract. The contract is to be put out for bids, and he believes that the lowest bid, excluding his own, can be represented by a distribution that is uniform between $8,000 and $20,000. Therefore, if the random variable X denotes the lowest of all other bids (in thousands of dollars), its probability density function is as follows:

a. What is the probability that the lowest of the other bids will be less than the consultant's cost estimate of $10,000?
b. If the consultant submits a bid of $12,000, what is the probability that he will secure the contract?
c. The consultant decides to submit a bid of $12,000.
What is his expected profit from this strategy?
d. If the consultant wants to submit a bid so that his expected profit is as high as possible, discuss how he should go about making this choice.

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