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# R to calculate the exact probability of 0 matches

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(r) To calculate the exact probability of 0 matches, divide the number of outcomes with 0 matches by the total number of possible outcomes. How does this result compare to your estimate from the simulation? P( X = 0) = Comparison:

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Chance/Rossman, 2015 ISCAM III Investigation B 16 (s) Use this method to determine the exact probabilities for each possible value for the number of matches. Express these probabilities as fractions and as decimals in the table below. Probability Distribution for # of correct matches: # matches 0 1 2 3 4 Probability (fraction) Probability (decimal) How do these theoretical probabilities compare to the empirical estimates you found in the simulation (question (i))? (t) What is the sum of these five probabilities? Why does this make sense? (u) What is the probability that at least one mother receives the correct baby? [ Hint : Determine this two different ways: first by adding the probabilities of the corresponding values, and then by taking one minus the probability that this event does not happen.] How does this compare to the simulation results? Probability rules: x The sum of the probabilities for all possible outcomes equals one. x Complement rule : The probability of an event happening is one minus the probability of the event not happening. x Addition rule for disjoint events : The probability of at least one of several events is the sum of the probabilities of those events as long as there are no outcomes in common across the events (i.e., the events are mutually exclusive or disjoint ). We can also consider the expected value of the number of matches, which is interpreted as the long-run average value of the random variable. For a discrete random variable we can calculate the expected value of the random variable X by again employing the idea of a weighted average of the different possible values of the random variable, but now the “weights” will be given by the probabilities of those values: E (X) = ¦ u values possible all value of y probabilit value ) ( ) ( (v) Calculate the expected value of the number of matches. Comment on how it compares to the average value you obtained in the simulation.
Chance/Rossman, 2015 ISCAM III Investigation B 17 (w) Is the expected value for the number of matches equal to the most probable outcome? If not, explain what is meant by an “expected” value. Notice that if we wanted to compute the average number of matches, say after 1000 trials, we would look at a weighted average: ¦ ¦ u ² u ² u ² u ² ² ² ² ² 1000 4 ) 4 (# 2 ) 2 (# 1 ) 1 (# 0 ) 0 (# ... 1000 0 2 1 0 1 s of s of s of s of x But from the results we saw above, each term (# of)/1000 terms converges to the probability of that outcome as we increase the number of repetitions, giving us the above formula for E ( X ). So we will interpret the expected value as the long-run average of the outcomes.

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