Lucy has a box of 30 chocolates. 18 are milk chocolate and the rest are dark chocolate. She takes a chocolate at random from the box and eats it. She then chooses a second. (a) Draw a tree diagram to show all the possible outcomes. (b) Calculate the probability that Lucy chooses: (i) 2 milk chocolates. (ii) A dark chocolate followed by a milk chocolate. Probability (Tree Diagrams) Conditional Events Milk Dark First Pick Second Pick Milk Dark Milk Dark

Aims: To understand and solve problems involving conditional probability.

 Name: Know what conditional events are.  Describe: The notation for conditional events A|B and how it differs from B|A and the influence of conditional probability on calculations with tree diagrams.  Explain: The probability laws regarding conditional events and use them to solve reverse conditional problems.  Evaluate: Whether an event is conditional or independent based on the above law.

 In the tree diagram question we have just seen the outcome of one event influenced the probability of another.  Where one event (A) occurring influences the probability of another event (B) happening the events are conditional (rather than independent).  We write the event that B happens when A is true as (B|A) read “B given A”.  P(B|A) is the probability that B is true given that A is true.

 We have already seen that for two successive events…  P(A  B) = P(A) x P(B)  This is only true where A and B are independent.  Where A and B are conditional…  P(A  B) = P(A) x P(B|A)

Given that the first counter picked is yellow what is the probability the second counter is…

Looking at the statistics of a basketball player’s free throws they have found that the probability of him scoring the first is 0.9 and if he scores the first the probability he scores the second is If he misses the first the probability he scores the second is 0.7. Find P(Scored 1 st |Scored 2 nd )

a)One of each colour was picked. b)That the first counter was red given that one of each colour was picked. There are four socks in Cliff’s drawer that are either white or black. If he takes two of them at random the probability he picks a pair of black socks is 0.5. How many socks of each colour are there?

 We have seen that P(A  B) = P(A) x P(B) is only true for independent events.  The law must be amended to the improved P(A  B) = P(A) x P(B|A) where A happening changes the probability of B happening.  This gives two means of deciding if A and B are independent…  If P(A  B) = P(A) x P(B) then A and B are independent.  If P(B) = P(B|A) then A and B are independent.

Given that the first counter picked is yellow what is the probability the second counter is…

a)One of each colour was picked. b)That the first counter was red given that one of each colour was picked. There are four socks in Cliff’s drawer that are either white or black. If he takes two of them at random the probability he picks a pair of black socks is 0.5. How many socks of each colour are there?