Download
1 / 21
210 likes | 223 Views
Learn about probability using concepts of independent and dependent events, as well as how to calculate probabilities of outcomes. Explore complement events and mutually exclusive events, and practice with examples.
E N D
Independent Events • Two events are Independent if the occurrence of 1 has no effect on the occurrence of the other. (a coin tossed 2 times, the first toss has no effect on the 2nd toss) • If A & B are independent events then the probability that both A & B occur is: • P(A and B) = P(A) • P(B)
A number cube is rolled and a coin is tossed. Find the probabilities: • P(5) • P(heads) • P(5 and heads)
P(5) = 1/6 • P(heads) = 1/2 • P(5 and heads) = P(5) P(heads) = =
Dependent Events • Two events A and B are dependent events if the occurrence of one affects the occurrence of the other.
Dependent Events • If A & B are dependent events, then the probability that both A & B occur is: • P(A&B) = P(A) * P(B/A) • The probability that B will occur given that A has occurred is called the conditional probability of B given A and is written P(B|A).
Comparing Dependent and Independent Events • You randomly select two cards from a standard 52-card deck. What is the probability that the first card is not a face card (a king, queen, or jack) and the second card is a face card if • (1) you replace the first card before selecting the second, and • (2) you do not replace the first card?
(1) If you replace the first card before selecting the second card, then A and B are independent events. So, the probability is: • P(A and B) = P(A) • P(B) = 40 * 12 = 30 52 52 169 • ≈ 0.178 • (2) If you do not replace the first card before selecting the second card, then A and B are dependent events. So, the probability is: • P(A and B) = P(A) • P(B|A) = 40*12 = 40 52 51 221 • ≈ .0181
To find P(A or B) you must consider what outcomes, if any, are in the intersection of A and B. • If there are none, then A and B are mutually exclusive events and P(A or B) = P(A)+P(B) • If A and B are not mutually exclusive, then the outcomes in the intersection (A and B) are counted twice when P(A) and P(B) are added. • So P(A and B) must be subtracted once from the sum
EXAMPLE 1 • One six-sided die is rolled. • What is the probability of rolling a multiple of 3 or 5? • P(A or B) = P(A) + P(B) = 2/6 + 1/6 = 1/2 • 0.5
EXAMPLE 2 • One six-sided die is rolled. What is the probability of rolling a multiple of 3 or a multiple of 2? • A = Mult 3 = 2 outcomes (3,6) • B = mult 2 = 3 outcomes (2,4,6) • P(A or B) = P(A) + P(B) – P(A and B) • P(A or B) = 2/6 + 3/6 – 1/6 = • 2/3 ≈ 0.67
EXAMPLE 3 • In a poll of high school juniors, 6 boys took French and 8 girls took french,11 boys took math class and 7 girls took math. • How many juniors surveyed were either girl or took math?
A = girl • B = took math • P(A) = 15/32, P(B) = 18/32 • P(A or B) = P(A) + P(B) – P(A and B) • P(A or B) = 15/32 + 18/32 – 7/32 = 26/32 = 13/16
Using complements to find Probability • The event A’, called the complement of event A, consists of all outcomes that are not in A. • The notation A’ is read ‘A prime’.
Probability of the complement of an event • The probability of the complement of A is : • P(A’) = 1 - P(A)
EXAMPLE 4 • A card is randomly selected from a standard deck of 52 cards. • Find the probability of the given event. • a. The card is not a king. 1 – P(king) = 1 – 4/52 = 48/52 ≈ 0.923
b. The card is not an ace or a jack. • P(not ace or jack) • 1 – P(ace or jack)= 1- P(4/52 + 4/52) = 1- 8/52 = 44/52 ≈ 0.846
In a survey of 200 pet owners, 103 owned dogs, 88 owned cats, 25 owned birds, and 18 owned reptiles. • 1. None of the respondents owned both a cat and a bird. • What is the probability that they owned a cat or a bird? • 113/200 • = 0.565 • 2. Of the respondents, 52 owned both a cat and a dog. • What is the probability that a respondent owned a cat or a dog? • 139/200 • = 0.695
Assignment Worksheet on Probability #’s 1-24
