Conditional Probability Unit 12 Lesson 2

CONDITIONAL PROBABILITY Students will be able to: find the conditional probability of an event. Key Vocabulary: Independent events Dependent events Conditional Probability

CONDITIONAL PROBABILITY Two events, 𝑨 and 𝑩, are INDEPENDENT EVENTS if the fact that event 𝑨 occurs does not affect the probability of 𝑩 occurring. Rolling a die and getting a 4, and then rolling a second die and getting a 5. Drawing a card from a deck and getting a jack, replacing it, and then drawing a second card and getting an ace.

CONDITIONAL PROBABILITY When the outcome or occurrence of the first event 𝑨 affects the outcome or occurrence of the second event 𝑩 in such a way that the probability is changed, the events are said to be DEPENDENT EVENTS. Drawing a card from a deck, not replacing it, and then drawing a second card. Having a high grades and getting a scholarship. Selecting a ball from a jar, not replacing it, and then selecting a second ball.

CONDITIONAL PROBABILITY A CONDITIONAL PROBABILITY of an event is used when the probability is affected by the knowledge of other circumstances. It can only occur to dependent events.

CONDITIONAL PROBABILITY A CONDITIONAL PROBABILITY of an event is used when the probability is affected by the knowledge of other circumstances. It can only occur to dependent events.

CONDITIONAL PROBABILITY The conditional probability of that the second event 𝑩 occurs given that the first event 𝑨 has occurred can be found by dividing the probability that both events occurred by the probability that the first event has occurred. It is denoted by 𝑷 𝑩|𝑨 , and is given by 𝑷 𝑩|𝑨 = 𝑷 𝑨 𝒂𝒏𝒅 𝑩 𝑷 𝑨 where 𝑷 𝑨 ≠𝟎.

CONDITIONAL PROBABILITY Sample Problem 1: Black and white chips are placed in a box. Jack selects two chips without replacing the first one. If the probability of selecting a black chip and white chip is 30/67, and the probability of selecting a white chip on the first draw is 6/11, find the probability of selecting a black chip on the second draw, given that the first chip is white.

CONDITIONAL PROBABILITY Sample Problem 1: Black and white chips are placed in a box. Jack selects two chips without replacing the first one. If the probability of selecting a black chip and white chip is 30/67, and the probability of selecting a white chip on the first draw is 6/11, find the probability of selecting a black chip on the second draw, given that the first chip is white. 𝑷 𝑾 = 𝟔 𝟏𝟏 𝑷 𝑾 𝒂𝒏𝒅 𝑩 = 𝟑𝟎 𝟔𝟕 𝑷 𝑩|𝑾 = 𝑷 𝑾 𝒂𝒏𝒅 𝑩 𝑷 𝑾 = 𝟑𝟎 𝟔𝟕 𝟔 𝟏𝟏 = 𝟑𝟎 𝟔𝟕 ⋅ 𝟏𝟏 𝟔 =𝑷 𝑩|𝑾 = 𝟓𝟓 𝟔𝟕 ≅𝟖𝟐.𝟎𝟗%

CONDITIONAL PROBABILITY Sample Problem 2: 9 red balls and 3 green marbles are place in a bag. Find the probability of randomly selecting a red marble on the first draw and a green marble on the second draw.

CONDITIONAL PROBABILITY Sample Problem 2: 9 red balls and 3 green marbles are place in a bag. Find the probability of randomly selecting a red marble on the first draw and a green marble on the second draw. 𝑷 𝑹 = 𝟗 𝟏𝟐 = 𝟑 𝟒 𝑷 𝑹 𝒂𝒏𝒅 𝑮 = 𝟑 𝟒 ⋅ 𝟑 𝟏𝟏 = 𝟗 𝟒𝟒 𝑷 𝑮|𝑹 = 𝑷 𝑹 𝒂𝒏𝒅 𝑮 𝑷 𝑹 = 𝟗 𝟒𝟒 𝟑 𝟒 = 𝟗 𝟒𝟒 ⋅ 𝟒 𝟑 𝑷 𝑮|𝑹 = 𝟑 𝟏𝟏 ≅𝟐𝟕.𝟐𝟕%

CONDITIONAL PROBABILITY Sample Problem 3: The probability Nathan buys butter is 0.10, and the probability that he buys bread and butter is 0.20. Find the probability that he will buy butter, given that he already bought bread.

CONDITIONAL PROBABILITY Sample Problem 3: The probability Nathan buys butter is 0.10, and the probability that he buys bread and butter is 0.20. Find the probability that he will buy butter, given that he already bought bread. 𝑷 𝒃𝒖𝒕𝒕𝒆𝒓 =𝟎.𝟑𝟕 𝑷 𝒃𝒓𝒆𝒂𝒅 𝒂𝒏𝒅 𝒃𝒖𝒕𝒕𝒆𝒓 =𝟎.𝟐𝟐 𝑷 𝒃𝒖𝒕𝒕𝒆𝒓|𝒃𝒓𝒆𝒂𝒅 = 𝑷 𝒃𝒓𝒆𝒂𝒅 𝒂𝒏𝒅 𝒃𝒖𝒕𝒕𝒆𝒓 𝑷 𝒃𝒖𝒕𝒕𝒆𝒓 = 𝟎.𝟐𝟐 𝟎.𝟑𝟕 =𝟎.𝟓𝟗𝟒𝟔 𝑷 𝒃𝒖𝒕𝒕𝒆𝒓|𝒃𝒓𝒆𝒂𝒅 =𝟓𝟗.𝟒𝟔%