Basics of Probability Some Definitions and Rules For

Basics of Probability Some Definitions and Rules

• For any random phenomenon, each attempt, or trial, generates an outcome – Ex: If the random phenomenon is flipping one coin, each flip is a trial, and the two possible outcomes are {heads} and {tails} – Ex: If the random phenomenon is rolling a die, each roll is a trial, and the six possible outcomes are {1}, {2}, {3}, {4}, {5}, {6}

• The outcomes of a random experiment must be: – exhaustive – all possible outcomes are included – mutually exclusive – no two outcomes can occur at the same time • Ex: You can’t roll a 5 and a 6 on the same roll of the die

• An event is an outcome or combination of outcomes – Ex: If we roll a single die, possible events include {4}, {number greater than 4}, {even number}

• The probability of an event is a measure describing how likely that particular outcome is to happen – P(A) means “probability of event A occurring” – Ex: In rolling a die, P(roll a 5) = 1/6 – Ex: In rolling a die, P(roll an even number) = 3/6 = ½ – Ex: In flipping a coin, P(heads) = ½ • What does P(heads) = ½ mean?

• Two events are independent if the outcome of one doesn’t influence or change the outcome of the other • Are the following events independent? – We flip a coin several times – We roll a die several times – We draw cards from a deck without replacing them

• The multiplication rule in probability says that for two independent events A and B, the probability that both occur is the product of the probabilities of the two events – We can write the formula as P(A and B) = P(A) * P(B) – Ex: The probability of rolling a 5 on a die roll is 1/6. If we roll a die twice, the probability of rolling a 5 on the first roll AND a 5 on the second roll is 1/6 * 1/6 = 1/36

• The addition rule in probability says that for any two events A and B, the probability that either occurs is the sum of the two probabilities minus the probability that both events occur together • We can write the formula as P(A or B) = P(A) + P(B) - P(A and B)

• In the case that the events are mutually exclusive (i. e. , both events cannot occur simultaneously) there is nothing to subtract • Ex: On one die roll, you cannot roll a 3 AND roll a 4, so the probability that either occurs is just the sum of the two probabilities • P(roll a 3 or roll a 4) = P(roll a 3) + P(roll a 4), since P(roll a 3 AND roll a 4) = 0 • But in the case that both events can occur simultaneously, you must subtract the probability of both events happening simultaneously

• Ex: On one die roll, you can roll a number greater than 4 AND roll an odd number, since 5 is both greater than 4 AND an odd number • P(roll a number greater than 4 or roll an odd number) = P(roll a number greater than 4) + P(roll an odd number) - P(roll a number greater than 4 AND an odd number) – P(roll a number greater than 4) = 2/6 – P(roll an odd number) = 3/6 – P(roll a number greater than 4 AND an odd number) = 1/6 • The only time that both events occur simultaneously is if you roll a 5 • P(roll a number greater than 4 or roll an odd number) = 2/6 + 3/6 - 1/6 = 4/6 • This makes sense because your condition is satisfied if you roll a 1, 3, 5, or 6, which represents 4 of the 6 outcomes on a single roll of the die

Practice Problems Assume that for a given baseball player, for each at bat, the probability of getting a hit is an independent event. Explain in your own words what it means for each at bat to be an independent event. It means that the probability of getting a hit in any given at bat is not influenced by what happens in any other at bat If the probability of getting a hit in a given at bat is 1/4, what is the probability that a player gets 4 hits in 4 at bats? P(hit and hit) = P(hit) * P(hit) = ¼ * ¼ * ¼ = 1/256 ≈ 0. 4%

Practice Problems In a class of 20 students, 12 are boys and 8 are girls. On a test, 2 boys and 2 girls failed. If a student from the class is randomly chosen, what is the probability of choosing a boy or a student who failed? P(boy or failed) = P(boy) + P(failed) - P(boy and failed) = 12/20 + 4/20 - 2/20 = 14/20

Practice Problems A day of the week is chosen at random. What is the probability of choosing Monday or Tuesday? P(Mon or Tue) = P(Mon) + P(Tue) = 1/7 + 1/7 = 2/7 Note that because choosing Monday and choosing Tuesday are independent events, there is nothing to subtract

Practice Problems 30% of Hunter students have taken ECON 100 and 25% of Hunter students have taken ENG 100. If half of Hunter students have taken either ECON 100 or ENG 100, what percent of Hunter students have taken both ECON 100 and ENG 100? P(ECON 100 or ENG 100) = P(ECON 100) + P(ENG 100) - P(ECON 100 and ENG 100) 50/100 = 30/100 + 25/100 - P(ECON 100 and ENG 100) = 5/100, or 5%