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Events are mutually exclusive if they cannot happen at the same time. For example, if we toss a coin, either heads or tails might turn up, but not heads and tails at the same time. Similarly, in a single throw of a die, we can only have one number shown at the top face. The numbers on the face are mutually exclusive events
If A and B are mutually exclusive events then the probability of A happening OR the probability of B happening is P(A) + P(B). • P(A or B) = P(A) + P(B)
Example 1 • What is the probability of a die showing a 2 or a 5?
Practice • The probabilities of three teams A, B and C winning a badminton competition are • Calculate the probability that • a) either A or B will win • b) either A or B or C will win • c) none of these teams will win • d) neither A nor B will win
Solution/s c) P(none will win) = 1 – P(A or B or C will win) d) P(neither A nor B will win) = 1 – P(either A or B will win)
Independent Events • Events are independent if the outcome of one event does not affect the outcome of another. For example, if you throw a die and a coin, the number on the die does not affect whether the result you get on the coin. • If A and B are independent events, then the probability of A happening AND the probability of B happening is P(A) × P(B). • P(A and B) = P(A) × P(B)
Example 1 • If a dice is thrown twice, find the probability of getting two 5’s.
Two sets of cards with a letter on each card as follows are placed into separate bags. Sara randomly picked one card from each bag. Find the probability that: a) She picked the letters ‘J’ and ‘R’. b) Both letters are ‘L’. c) Both letters are vowels.
Solution for no. 2 • Probability that she picked J and R = • Probability that both letters are L = • Probability that both letters are vowels =
Example 3 • Two fair dice, one colored white and one colored red, are thrown. Find the probability that: • a) the score on the red die is 2 and white die is 5. • b) the score on the white die is 1 and red die is even
Solution for No. 3 • Probability the red die shows 2 and white die 5 = • Probability the white die shows 1 and red die shows an even number =
DEPENDENT EVENTS • Events are dependent if the outcome of one event affects the outcome of another. For example, if you draw two colored balls from a bag and the first ball is not replaced before you draw the second ball then the outcome of the second draw will be affected by the outcome of the first draw.
If A and B are dependent events, then the probability of A happening AND the probability of B happening, given A, is P(A) × P(B after A). • P(A and B) = P(A) × P(B after A) • P(B after A) can also be written as P(B | A) • then P(A and B) = P(A) × P(B | A)
Example 1 • A purse contains four P50 bills, five P100 bills and three P20 bills. Two bills are selected without the first selection being replaced. Find P(P50, then P50)
There are four P50 bills. • There are a total of twelve bills. • P(P50) = 4/12 • The result of the first draw affected the probability of the second draw. • There are three P50 bills left. • There are a total of eleven bills left. • P(P50 after P50) = 3/11
P(P50, then P50) = P(P50) · P(P50 after P50) = (4/12)x(3/11)=12/132 • The probability of drawing a P50 bill and then a P50bill is
Dependent:Practice • A bag contains 6 red, 5 blue and 4 yellow marbles. Two marbles are drawn, but the first marble drawn is not replaced. • a) Find P(red, then blue). • b) Find P(blue, then blue)
Independent Events: Practice • Two fair dice, one colored white and one colored red, are thrown. Find the probability that: • a) the score on the red die is 2 and white die is 5. • b) the score on the white die is 1 and red die is even
Mutually ExclusiveEvents:Practice • The probabilities of three teams A, B and C winning a badminton competition are • Calculate the probability that • a) either A or B will win • b) either A or B or C will win • c) none of these teams will win • d) neither A nor B will win
