P(A and B) = P(A) x P(B) The ‘AND’ Rule The Probability of events A AND B happening: P(A and B) = P(A) x P(B)

I flip two coins, what is the probability that I get two heads? P(HH) = 1 4 Coin 1 Coin 2 1 2 Heads P(HH) = x = 1 2 1 2 1 4 Heads 1 2 P(HT) = x = 1 2 1 2 1 4 1 2 Tails 1 2 Heads P(TH) = x = 1 2 4 1 2 Tails P(TT) = x = 1 2 4 1 2 Probabilities on each branch add up to 1 Tails Possible outcomes are shown on the end of the branches Multiply along the branches to work out the probabilities

Stops 0.2 Stops Doesn’t 0.3 Stop 0.8 Stops 0.2 0.7 Doesn’t Stop 0.8 Helen passes through 2 sets of traffic lights on her way to work. The probability that she stops at the first set is 0.3 The probability that she stops at the second set is 0.2 What is the probability that she has to stop exactly once? = 0.24 + 0.14 = 0.38 Lights 1 Lights 2 Stops P(SS) = 0.3 x 0.2 = 0.06 0.2 Stops Doesn’t Stop 0.3 P(SD) = 0.3 x 0.8 = 0.24 0.8 Stops P(DS) = 0.7 x 0.2 = 0.14 0.2 0.7 Doesn’t Stop 0.8 Doesn’t Stop P(DD) = 0.7 x 0.8 = 0.56

I have a bag of sweets, 3 blue and 7 red I have a bag of sweets, 3 blue and 7 red. I pick one sweet out without looking, eat it and then pick out and eat a second sweet. What is the probability that I eat one blue and one red sweet? 42 90 7 15 Sweet 1 Sweet 2 = 6 9 Red P(RR) = x = 7 10 6 9 42 90 Red 7 10 3 9 P(RB) = x = 7 10 3 9 21 90 Blue Plenary 7 9 Red P(BR) = x = 3 10 7 9 21 90 3 10 Blue P(BB) = x = 3 10 2 9 6 90 2 9 Blue

We can solve this problem by drawing a tree diagram. Tuesday Look! Vertically, the numbers add up to 1. It rains 0.3 Monday We can solve this problem by drawing a tree diagram. We know that the probability is 0.2 It rains Look! Vertically, the numbers add up to 1. 0.2 There are two possible events here; It rains or It does not rain 0.7 It does not rain Now let’s look at Tuesday The probability that it rains on Tuesday was given to us as 0.3. We can work out that the probability of it not raining has to be 0.7, because they have to add up to 1. Look! Vertically, the numbers add up to 1. It rains 0.3 0.8 It does not rain Well, raining and not raining are Mutually Exclusive Events. So their probabilities have to add up to 1. What is the probability? 1 – 0.2 = 0.8 We now have the required Tree Diagram. 0.7 It does not rain

So the probability that it rains on Monday and Tuesday is 0.06 We wanted to know the probability that it rained on Monday and Tuesday. Monday It rains It does not rain Tuesday 0.3 0.7 0.2 0.8 0.2 x 0.3 = 0.06 This is the only path through the tree which gives us rain on both days We can work out the probability of both events happening by multiplying the individual probabilities together So the probability that it rains on Monday and Tuesday is 0.06

Actually, we can work out the probabilities of all the possible events Monday It rains It does not rain Tuesday 0.3 0.7 0.2 0.8 0.2 x 0.7 = 0.14 The probability of rain on Monday, but no rain on Tuesday is 0.14

Monday It rains It does not rain Tuesday 0.3 0.7 0.2 0.8 0.8 x 0.3 = 0.24 The probability that it will not rain on Monday, but will rain on Tuesday is 0.24

Monday It rains It does not rain Tuesday 0.3 0.7 0.2 0.8 0.8 x 0.7 = 0.56 The probability that it does not rain on both days is 0.56

Let’s look at the completed tree diagram Monday It rains It does not rain Tuesday 0.3 0.7 0.2 0.8 0.06 0.14 0.24 0.56 The end probabilities add up to 1. Remember this! It can help you check your answer! What do you notice?

There are 2 sets of traffic lights on a journey There are 2 sets of traffic lights on a journey. A car approaches the first set where the probability of stopping is 0.7. The car then gets to the second set where the probability of stopping is 0.4 Draw a tree diagram Find the probability of: Stopping at both sets of lights Only stopping at one set Julie and Pat are going to the cinema. The probability that Julie will arrive late is 0.2. The probability that Pat will arrive late is 0.6. The two events are independent. Draw the tree diagram for this data Find the probability that: both will arrive late neither will arrive late

Julie and Pat are going to the cinema Julie and Pat are going to the cinema. The probability that Julie will arrive late is 0.2. The probability that Pat will arrive late is 0.6. The two events are independent. Draw the tree diagram for this data Find the probability that: both will arrive late neither will arrive late Salika travels to school by train every day. The probability that her train will be late on any day is 0.3 Draw the probability tree diagram for Monday and Tuesday. (b) Work out the probability that she is not late on either day Work out the probability that she is late on at least one of these days

Two balls are chosen from a bag WITHOUT REPLACEMENT Two balls are chosen from a bag WITHOUT REPLACEMENT. The bag contains 5 red and 9 blue balls. Draw the tree diagram Find the probability of choosing two red balls Find the probability of choosing at least one blue ball.

Salika travels to school by train every day. The probability that her train will be late on any day is 0.3 Draw the probability tree diagram for Monday and Tuesday. (b) Work out the probability that she is not late on either day Work out the probability that she is late on at least one of these days 4. Jacob has 2 bags of sweets. Bag P contains 3 green sweets and 4 red sweets. Bag Q contains 1 green sweet and 3 yellow sweets. Jacob takes one sweet at random from each bag. Draw the tree diagram for each bag Work out the probability of picking two green sweets Work out the probability of getting at least at least one green sweet

4. Jacob has 2 bags of sweets. Bag P contains 3 green sweets and 4 red sweets. Bag Q contains 1 green sweet and 3 yellow sweets. Jacob takes one sweet at random from each bag. Draw the tree diagram for each bag Work out the probability of picking two green sweets Work out the probability of getting at least at least one green sweet 5. There are 12 girls and 8 boys in a group. Two are chosen at random by producing names out of a hat. The chosen two are then required to give a presentation to the rest of the group. (a) Draw a tree diagram (b) Find the probability of choosing (i) 2 girls (ii) 1 boy and 1 girl in any order