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STARTER. Given this table for the probability distribution of X, calculate E(X) A random variable X can take any one of the value 0, 1 and 2. The probability P(X = x) is given by: x 2 + x + 4 20 Calculate E(X). E(X) = 3.9. E(X) = 1.3. Note 3 : Expected Value Applications.

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  1. STARTER Given this table for the probability distribution of X, calculate E(X) A random variable X can take any one of the value 0, 1 and 2. The probability P(X = x) is given by: x2 + x + 4 20 Calculate E(X) E(X) = 3.9 E(X) = 1.3

  2. Note 3 : Expected Value Applications In a game of chance, a game is said to be fair if the entry fee is equal to the expected winnings. Expected return = expected winnings – entry price = 0 in a fair game

  3. Example: A raffle has prizes $20, $10 or $1 with probabilities 0.01, 0.02 and 0.02 respectively. What should the price of each ticket be, for the raffle to be ‘fair’. E(W) = 0.01(20 - t) + 0.02(10 - t) + 0.02(1 - t) -0.95t For a fair game expected value = 0 0 = 0.2 – 0.01t + 0.2 – 0.02t + 0.02 – 0.02t – 0.95t 0 = 0.42 - t Ticket price would be 42 cents

  4. Note 3: Binomial Distribution Any experiment in which there are only two outcomes (success or failure) is called a Bernoulli trial. The conditions for X to have a binomial distribution are: • There are a fixed number of identical trials (n) • There are only two possible outcomes for each trial – a ‘success’ or a ‘failure’ • Probability of success (p) at each trial must be constant • Each trial is independent of the other trials

  5. There are two parameters n and p. • The random variable X is the total number of successes in n trials • p is the probability of success at an individual trial • Any binomial distribution is represented as The probability of obtaining r successes out of n independent trials, when p is the probability of success for each trial is:

  6. Example: A duck shooter has a probability of 0.4 of hitting any duck that he shoots at during a hunt. In a hunt he fires 10 shots. Explain why the duck shooter can be modelled by a binomial distribution: • There are a fixed number of identical trials There are 10 shots • There are only two possible outcomes for each trial Each shot – hit or miss • Probability of success at each trial must constant Probability of hitting a duck remains constant at 0.4 • Each trial is independent of the other trials Each shot is independent of the other

  7. Find the probability that he shoots exactly 8 ducks n = 10 p = 0.4 r = 8 P(X = 8) = = 0.01062

  8. Find the probability that he shoots exactly 8 ducks (CALCULATOR) STAT DIST BINM Bpd Var x 8 Numtrial 10 p 0.4 Execute P(X = 8) = 0.01062

  9. New Book Page 531 Exercise 15C

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