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Discrete Mathematics CS 2610

Discrete Mathematics CS 2610. November 5, 2008. Probability. We will be focusing on discrete probability. Specifically, we focus on probabilities involving a finite number of equally probable outcomes.

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Discrete Mathematics CS 2610

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  1. Discrete Mathematics CS 2610 November 5, 2008

  2. Probability • We will be focusing on discrete probability. Specifically, we focus on probabilities involving a finite number of equally probable outcomes. • From this perspective, we can define the probability of an event occurring as the number of successful outcomes divided by the number of possible outcomes.

  3. Probability We can understand probability by considering sets of outcomes: We define a set S to be a sample space, a set of all possible outcomes of some experiment. We define a set E  S, the set of all outcomes in which the event occurs. We further assume that all outcomes in S are equally likely. Then the probability of the event occurring is: p(E) = |E| / |S|

  4. Probability – basic problems • A bag contains 8 grape jellybeans and 6 raspberry jellybeans. What is the probability of selecting a grape jellybean? A raspberry jellybean? • You roll two six-sided dice. What is the probability of a pair? Of a sum of 7? Of a sum of 3? • You are dealt a 5-card poker hand. What is the probability of a royal flush? Of 4-of-a-kind? Of a pair of jacks? A full house?

  5. Probability • We use p(E) to denote the probability that an event occurs. • We use p(E) to denote the probability that an event does not occur. P(E) = 1 – p(E) If a coin is flipped 5 times, what is the probability of at least one head coming up?

  6. Probability • If E1 and E2 are two events in the same sample space, then p(E1 E2) = p(E1) + p(E2) – p(E1  E2) • It’s just the subtraction principle again! A number is selected at random from the set of positive integers less than or equal to 100. What is the probability the number is divisible by either 2 or 5?

  7. Probability One of the most interesting and (for some) counterintuitive applications of discrete probability is the Monty Hall problem. • There are 3 doors, only one of which has a prize behind it. • After selecting a door, the contestant is shown one of the doors not picked that doesn’t contain a prize. • The contestant can then keep their door, or switching to the other unopened door. What should the contestant choose?

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