Tree Diagram with Conditional Probabilities Notes 16 Conditional

Tree Diagram with Conditional Probabilities Notes #16

Conditional Probability contains a condition that may limit the sample space for an event. You can write a conditional probability using the notation This reads “The Probability of Event B, given Event A”

Ex 1: Constructing a tree diagram This coming Friday it could start out 0. 28 chance of clear skies or 0. 72 cloudy. Given that it is a clear day, there is a 0. 04 chance of rain. However, if the day started out cloudy, then there is chance of 0. 31 chance of rain. Construct a Tree Diagram to show this information. Steps: 1) Identify the Events Precipitation 2) Note the values for each condition. Skies P(C) = 0. 28 P(C’) = 0. 72 P( R | C ) = 0. 04 Rain R: Rain R’: No Rain Clear P( R’ | C ) = 0. 96 No Rain P( R | C’ ) = 0. 31 Rain Cloudy P( R’ | C’ ) = 0. 69 Key C: Clear C’ : Cloudy No Rain

Key C: Clear C’ : Cloudy R: Rain R’: No Rain a. What is the probability that a day will start out clear, and then will rain? The path containing CLEAR AND RAIN represents DAYS THAT START OUT CLEAR and then will rain. P(C∩R)= P(C) • P( R | C ) = 0. 28 • 0. 04 = 0. 011 The probability that a day will start out clear and then rain is about 0. 011 or 1%.

Key C: Clear C’ : Cloudy 0. 28 x 0. 96 = 0. 2688 0. 72 x 0. 69 = 0. 4968 b. Find the probability that it will not rain on any given day. The paths containing CLEAR AND NO RAIN and CLOUDY AND NO RAIN both represent a day when it will not rain. Find the probability for both paths and add them. = P(C ∩ R’) + P(C’ ∩ R’) = [P(C) • P(R’ | C)] + [P(C’) • P(R’ | C’)] = 0. 28(. 96) +. 72(. 69) = 0. 7656 ANS: The probability that it will not rain on any given day is about 77%. R: Rain R’: No Rain

Constructing a tree diagram Construct a tree diagram to show this information. Tuesday Rain Monday Rain No rain *Keep the denominators the same in the final ANS till you change it to a decimal. * Rain No rain = P(R ∩ R) = P(R ∩ R’) = P(R’ ∩ R’)

Probabilities =. 48 =. 12 =. 24 =. 16 Calculate a) P(it rains at least once) b) P(it rains one day only) c) it rains on one day only, given it rains at least once. Or 0. 84 Or 0. 36