Binomial Probability Distribution

Binomial Probability: In a binomial experiment there are two mutually exclusive  outcomes, often referred to as "success" and "failure".  The probability of success is p. The probability of failure is 1 – p. These probability experiments are sometimes referred to as a Bernoulli trial, after Swiss mathematician Jacob Bernoulli, (1654 - 1705).

The Probability of EXACTLY: When computing a binomial probability, it is necessary to calculate and multiply three separate factors: 1.  The number of ways to select exactly r successes 2.  The probability of success (p) raised to the r power 3.  The probability of failure (q) raised to the (n - r) power.

The probability of an event, p, occurring exactly r times: n = number of trials  r = number of specific events you wish to        obtain  p = probability that the event will occur  q = probability that the event will not occur    (q = 1 - p, the complement of the event)

Ex (1) - A test consists of 10 multiple choice questions with five choices for each question.  As an experiment, you GUESS on each and every answer without even reading the questions.  What is the probability of getting exactly 6 questions correct on this test? (answers rounded to three decimal places)

Using your TI – 83 to get answer: Find the built-in command binompdf (binomial probability density function), it can be used to quickly determine "exactly“ problems. Hit the DISTR (2nd VARS) button. Click down to A: binompdf( Type in: binompdf (number of trials, probability of occurrence, number of specific events)

Ex (2) - When rolling a die 100 times, what is the probability of rolling a "4" exactly 25 times? (answers rounded to three decimal places)

Ex (3) - At a certain intersection, the light for eastbound traffic is red for 15 seconds, yellow for 5 seconds, and green for 30 seconds.  Find the probability that out of the next eight eastbound cars that arrive randomly at the light, exactly three will be stopped by a red light. (answers rounded to three decimal places)

Ex (4) - The probability that Bob will score above a 90 on a statistics test is 4/5.  What is the probability that he will score above a 90 on exactly three of the four tests this quarter? (answers rounded to three decimal places)

Ex (5) - Which fraction represents the probability of obtaining exactly eight heads in ten tosses of a fair coin? (a) 45/1024 (b) 64/1024 (c) 90/1024 (d) 180/1024

“At least” or “At most” When computing "at least" and "at most" probabilities, it is necessary to consider, in addition to the given probability, all probabilities larger than the given probability ("at least") all probabilities smaller than the given probability ("at most")

Formula: n = number of trials  r = number of specific events you wish to obtain  p = probability that the event will occur  q = probability that the event will not occur    (q = 1 - p, the complement of the event)

Ex (6) - A bag contains 6 red marbles, 4 blue marbles, and 7 white marbles. What is the probability of drawing a red marble at least 3 out of 5 times? (answers rounded to three decimal places) * To solve this problem, we need to find the probabilities that r could be 3 or 4 or 5, to satisfy the condition "at least". It will be necessary to compute for r = 3, r = 4 and r = 5 and add these three probabilities for the final answer. We need to compute three times:

Ex (6) - Marbles work and answer: For r = 3: For r = 4: For r = 5: Sum: .184 + .050 + .005 = .239 rounded to the nearest hundredth = 0.24  ANSWER

Ex (7) - A family consists of 3 children Ex (7) - A family consists of 3 children.  What is the probability that at most 2 of the children are boys?  (answers rounded to three decimal places) * To solve this problem, we need to find the probabilities for "At most" 2 boys which implies that there could be 0, 1, or 2 boys. Three boys is not an option.

Ex (7) - Team A and Team B are playing in an eight-ball league Ex (7) - Team A and Team B are playing in an eight-ball league.  They will play each other five times.  If the probability that team A wins a game is 1/3, what is the probability that team A will win at least three of the five games? (answers rounded to three decimal places)

Ex (8) - On any given day, the probability that the entire Najuch family eats dinner together is 2/5.  Find the probability that, during any 7-day period, the Najuch's eat dinner together at least six times. (answers rounded to three decimal places)

TI – 83 for “At most” There is a built-in command binomcdf (binomial cumulative density function) that can be used to quickly determine "at most“ problems. Hit the DISTR (2nd VARS) button Choose B: binomcdf( Input: binomcdf (number of trials, probability of occurrence, number of specific events)