Learning Goal 13: Probability Use the basic laws of probability by finding the probabilities of mutually exclusive events. Find the probabilities of dependent events. Calculate conditional probabilities. Use expected value to predict outcomes. Probability

Learning Goal 13: Probability Use the basic laws of probability by finding the probabilities of mutually exclusive events. Find the probabilities of dependent events. Calculate conditional probabilities. Use expected value to predict outcomes. Expected Values (The Same as a Weighted Average): The average over the long run. The sum of all the possible outcomes times there probabilities.

Learning Goal 13: Probability Use the basic laws of probability by finding the probabilities of mutually exclusive events. Find the probabilities of dependent events. Calculate conditional probabilities. Use expected value to predict outcomes. Expected Values: Deal or No Deal Calculate the expected value at the start of the game and at the end of each round.

Learning Goal 13: Probability Use the basic laws of probability by finding the probabilities of mutually exclusive events. Find the probabilities of dependent events. Calculate conditional probabilities. Use expected value to predict outcomes. Sample space: the collection of all possible outcomes. Event: any collection of outcomes from the sample space. The Event is a subset of the sample space. Probability Model: Simple event: consists of exactly one outcome.

Learning Goal 13: Probability Use the basic laws of probability by finding the probabilities of mutually exclusive events. Find the probabilities of dependent events. Calculate conditional probabilities. Use expected value to predict outcomes. Probability

Learning Goal 13: Probability Use the basic laws of probability by finding the probabilities of mutually exclusive events. Find the probabilities of dependent events. Calculate conditional probabilities. Use expected value to predict outcomes. Probability Rules The probability P(A) of event A satisfies 0 < P(A) < 1 If S is the total sample space in a probability model, then P(S) = 1 Two events A and B are disjoint (mutually exclusive) if they have no outcomes in common [P(A and B) = 0] and so can never occur simultaneously. If A and B are disjoint, –P(A or B) = P(A) + P(B) –P(A and B) = 0 The complement of any event A is the event that A does not occur. The complement rule states that –P(A c ) = 1 - P(A)

Learning Goal 13: Probability Use the basic laws of probability by finding the probabilities of mutually exclusive events. Find the probabilities of dependent events. Calculate conditional probabilities. Use expected value to predict outcomes. General Addition Rule for Unions of Two Events For any two events A and B –P(A or B) = P(A) + P(B) – P(A and B) Equivalently –P(A U B) = P(A) + P(B) – P(A ∩ B) Great place to see in a Venn diagram!!!

Learning Goal 13: Probability Use the basic laws of probability by finding the probabilities of mutually exclusive events. Find the probabilities of dependent events. Calculate conditional probabilities. Use expected value to predict outcomes. Rule 5: Multiplication Rule Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, P(A and B) = P(A)P(B) This is the multiplication rule for independent events

Learning Goal 13: Probability Use the basic laws of probability by finding the probabilities of mutually exclusive events. Find the probabilities of dependent events. Calculate conditional probabilities. Use expected value to predict outcomes. Spinner Learning Task 4

Learning Goal 13: Probability Use the basic laws of probability by finding the probabilities of mutually exclusive events. Find the probabilities of dependent events. Calculate conditional probabilities. Use expected value to predict outcomes. Monty Hall, Lets Make a Deal The game show had three “doors”. Behind one door was a very nice prize, like a new car and behind the other two doors were prizes like goats or camels. The contestant picks a door and the host, Monty Hall, opens one of the remaining doors, the one he knows doesn’t hide the car. The contestant is given the option to switch doors. What is the probability of winning the car if the contestant stays with there first choice? If the decide to switch?

Learning Goal 13: Probability Use the basic laws of probability by finding the probabilities of mutually exclusive events. Find the probabilities of dependent events. Calculate conditional probabilities. Use expected value to predict outcomes. Conditional Probability When P(A)>0, using ALGEBRA, the conditional probability of B, given A, is –P(B|A) = However, just think about it, if you are given information that A has occurred, the sample space has reduced, and we need to find P(B) in this reduced sample space.