Number of households with a cat Number of households without a cat Totals Number of households with a dog 1015 Number of house holds without a dog 2530 Totals Puzzle to think about! Complete the table above and try to answer these questions about households from the above survey: a)How many households where surveyed? b)What is the probability a household owns a cat? c)What is the probability a household owns a dog and a cat? d)What is the probability a household owns a cat given they own a dog? e)What is the probability a household owns a dog given they own a cat? f)What is the probability a household owns a dog given they don’t own a cat?

Aims: To know and be able to use the conditional rule: To practice exam style probability questions. Probability Lesson 2

Sometimes the probability of event B happening depends on whether A has happened or not. For example, if blue and green counters are pulled from a bag twice and not replaced, then the probability of pulling out a green counter on the second try, will depend on what colour was pulled out on the first try. Conditional probability The probability that event A will happen, given that event B has happened, is written P(A | B) This is a c_________________ probability.

Conditional probability B A A′ B B′ P(A) P(A′) P(B | A) P(B′ | A) P(B | A′ ) P(B′ | A′ ) P(A B) = P(A) × P(B | A) P(A B′) = P(A) × P(B′ | A) P(A′ B′) = P(A′) × P(B′ | A′) P(A′ B) = P(A′) × P(B | A′) To find the probability of events A and B both happening we use: P(A B) = P(A) × P(B | A) This formula can be re-arranged to give: Given in formula booklet

D M D D M M Example: A bag contains 8 dark chocolates and 4 milk chocolates. One chocolate is taken out and eaten. A second chocolate is then taken. Find the probability of a) two milk chocolates being taken b) both chocolates being of different types. Conditional probability a) P(M M) = b) P(D M) = P(M D) =

Examination style question: A man has 2 shirts (one white and one blue) and 2 ties (red and silver). If he wears the white shirt, he chooses the red tie with probability 0.4. If he wears the blue shirt, he chooses the red tie with probability The probability that he wears the white shirt is 0.7. Conditional probability a)Find the probability that he wears the red tie. b)Given that he is wearing a red tie, find the probability that he picked the blue shirt. W B R R S S Shirt Tie a)P(red tie) = P(W R) + P(B R) = (0.7 × 0.4) + (0.3 × 0.75) =

So, b)Recall the formula for conditional probability: Conditional probability = Example 2: A disease affects 1 in 500 people. A diagnostic test for the disease records a positive result 99% of the time when the disease is present (this is called the sensitivity of the test). The test records a negative result 95% of the time when the disease in not present. The test results are always either positive or negative. Find the probability that a person has the disease, given that the test result is positive. 5 mins have a go! Then answer on your w/b

D D′ +ve –ve DiseaseTest P(D +ve) = × 0.99 = P(D′ +ve) = × 0.05 = Conditional probability So, Exam relay questions in pairs or threes. Homework – Complete the Moodle Probability homework

Monty Hall problem A game show host offers you one of three doors: behind one door is a car, behind the others there are goats. He asks if you would like to change your choice to the other unopened door. Should you? You pick a door, and the host, who knows what is behind the doors, opens one of the others to reveal a goat.

Monty Hall problem