STATISTICS - CLUTCH CH.5: THE BINOMIAL RANDOM VARIABLE.

Transcription

1 !!

2 Probability BINOMIAL DISTRIBUTIONS Binomial distributions are a specific type of distribution They refer to events in which there are only two possible outcomes heads/tails, win/lose, pass/fail, right/wrong These two events are referred to as and x = number of successes n = number of trials p = probability of success q = probability of failure There are probabilities associated with both of these Success doesn t mean it s ; it s just which of the two outcomes you re interested in Suppose you really like flipping quarters so you flip 4 quarters and you want to see how many heads you end up with: p = 0.25 p = 0.50 p = Number of Heads MEAN AND SD The mean and standard deviation ( ) of the binomial distribution are simpler than those for the other DRVs. μ x = np σ x = npq EXAMPLE 1: Suppose you flip 6 quarters. What is the expected number of heads and the standard deviation for this distribution? EXAMPLE 2: You have no idea what your 100-question exam is about, but you sat in class a few times. It starts in 5 minutes. If there is a 60% chance that you d get a right answer, what is the average number of questions you ll most likely get correct? Provide a measure of spread for this average. Page 2

3 PRACTICE 1: A coin is tossed 100 times, what is the standard deviation of the binomial distribution for the number of heads you would end up with? PRACTICE 2: If there is a 30% chance of conceiving a boy, how many do you expect to have if you want to have 4 kids? PRACTICE 3: You re playing beer pong. You average a 30% miss rate. Out of 6 shots, how many do you expect to make? What is the standard deviation of this binomial distribution? PRACTICE 4: Pregnancy scares are serious. If there is a.01% chance that a condom breaks, what is the average number of condoms you d expect to break out of 100? What is the variance for this distribution? Page 3

4 COMBINATIONS - ncx scary right? Before we dive into the formula, let s break dow the com o e ts to get a better u dersta di g When looking at two events occurring simultaneously, there might multiple ways they can happen EXAMPLE 1: You flip a quarter four times. Write out all the possible ways you can end up with exactly two heads. It s easy to see that o ce more trials are thrown into the mix, it can get difficult thinking of all these possibilities The function can help determine all these potential ways two events can occur ncx = ( k ) = ( - ) x = number of successes n = number of trials EXAMPLE 2: You flip a quarter 6 times. How many different ways can you end up with exactly two heads? MULTIPLICATION RULE When finding probabilities for these independent events, remember that the multiplication rule can be applied P(A B C D ) = P(A) P(B) P(C) P(D) The probability of multiple events happening is simply the product of each of their EXAMPLE 3: You are taking a True/False quiz with 10 questions. What is the probability of guessing all of them correct? Remember that all possible ways events can occur together must be taken into account Page 4

5 BINOMIAL FORMULA The binomial formula is able to spit out all possible of events as well as the probabilities associated with them ncx p n- It s best to start by writing down all the given variables from the word problem x = number of successes n = number of trials p = probability of success q = probability of failure EXAMPLE 1: What is the probability of getting five True/False questions all correct by simply guessing? EXAMPLE 2: You flip a quarter five times. What is the probability that you flip exactly two heads? EXAMPLE 3: The probability of having a girl is 65%, on average. What is the probability that, out of seven kids, a randomly selected couple will pop out at least two girls? Page 5

6 PRACTICE 1: There s a 50% chance that the average student who applies to graduate school will actually get admitted. What is the probability that a student would get into 3 randomly selected grad schools out of 10? PRACTICE 2: Referring to Practice 1, what is the probability that a student would get into more than 8 grad schools? PRACTICE 3: Referring to Practice 1, what is the probability that a student would get into at most 9 grad schools? PRACTICE 4: The chance, on average, of going home alone after a night out is 70%. What is the probability that, on a random 15 nights out, you take someone home every time? PRACTICE 5: Referring to Practice 4, what is the probability of picking up at least two people? Page 6

7 BINOMIAL TABLES Once you have to calculate 3 or more binomial probabilities by hand, things can easily get out of hand Example: P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) The good thing is that your book has to take care of this nightmare Below is a sample of the table: P(X x) = P(X = 0) + P(X = 1) + + P(X = x) P(X x) = nc0p 0 q n + nc1p 1 q n ncxp x q n-x p x n = This table has probabilities, not probabilities for each instance Example: the highlighted number is the probability of at most 3 successes out of 5 with a 70% chance of success Another way of saying this is: = P(X 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) EXAMPLE 1: In a game of slots, the chance of you winning any money is 30% (including small wins and jackpots). What is the probability that if you will win money in at most 3 out of 5 games of slots? EXAMPLE 2: Referring to Example 1, what is the probability that you will win money in at least 2 out of 5 games of slots? EXAMPLE 3: Referring to Example 1, what is the probability that you will win money in exactly 4 out of 5 games of slots? Page 7

8 PRACTICE 1: Your sibling challenges you to a game of volleyball. In any given volley, you have a 20% chance of scoring. What is the probability that, in any 20 volleys, you score on 10 of them? PRACTICE 5: Your partner hates boys, but you want a few. Assuming that humans could produce 10 kids easily and there s a 70% chance of having a boy, what is the probability of having between 2 and 8 boys, inclusive? PRACTICE 2: Referring to Practice 1, what is the probability that you score on at least 3 of the volleys? PRACTICE 3: Referring to Practice 1, what is the probability that you score on more than 18 of the volleys? PRACTICE 6: Referring to Practice 5, what is the probability of having between 1 and 5 girls, not inclusive? PRACTICE 4: Referring to Practice 1, what is the probability that you score on less than 8 of the volleys? Page 8

9 P(X x) = P(X = 0) + P(X = 1) + + P(X = x) P(X x) = nc0p 0 q n + nc1p 1 q n ncxp x q n-x p x n = n = n = n = n = n = n = n = Page 9

10 p x n = n = n = n = Page 10

11 p x n = Page 11