Maths: Analysis and Approaches · Statistics and probability. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
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A discrete random variable X takes separate values, each with its own probability. A table of the values and their probabilities is the probability distribution, and the probabilities must add up to 1.
The expected value E(X) is the long-run average of X: multiply each value by its probability and add. A game is fair when the expected gain is zero, so the expected winnings equal the cost of playing.
The binomial distribution B(n, p) counts the number of successes in n trials. It fits only when the number of trials is fixed, each trial has two outcomes, the trials are independent, and the probability of success p stays the same.
Leaving out the nCr term in a binomial probability. pr(1 − p)n − r is the probability of only one order of successes and failures, so multiply by the number of orders.
Answer: P(X = 1) = 0.4116 (0.412 to 3 significant figures), mean 1.2, variance 0.84
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