## What you need to know

**Probability & Tree Diagrams**

Probability is the study of how likely things are to happen. This topic will look at how **tree diagrams**, can be used to determine the probability of different types of events happening.

Make sure you are happy with the following topics before continuing.

**Independent and Dependent Events**

We need to understand **independent** and **dependent** events to be able to do the next sections.

- Two or more events are
**independent**if one event**doesn’t**effect the probability of the others happening. - Two or more events are
**dependent**if one event**does**effect the probability of the others happening.

**Example:**

**Getting a head both times on 2 coin flips are independent events.****Picking a red marble then picking a green marble without replacing the red marble are dependent events.**

**The AND Rule**

- The
**AND rule**states that:**If two events, A and B, are independent, then,**

\textcolor{black}{\text{P}(A \text{ and } B) = \text{P}(A) \times \text{P}(B)}

**The OR Rule**

- The
**OR rule**states that:**for two events, A and B, then,**

\textcolor{black}{\text{P}(A \text{ or } B) = \text{P}(A) + \text{P}(B) - \text{P}(A \text{ and } B)}

**If A and B cannot happen together, we say they are mutually exclusive, and then we have \text{P}(A \text{ and } B) = 0, so the OR rule becomes**

\textcolor{black}{\text{P}(A \text{ or } B) = \text{P}(A) + \text{P}(B)}

**Probability Trees**

**Probability trees** are similar to frequency trees, but we instead put the probabilities on the branches and the events at the end of the branch.

**Example: **A bag contains 4 **red** balls and 5 **blue** balls. Raheem picks 2 balls at random. Calculate the probability that he selects the same coloured ball each time, given that after each time a ball is selected, **it is replaced**.

**Step 1:**Construct the probability tree showing two selections, noting that there are 9 balls in total and a \dfrac{4}{9} chance of picking a**red**ball. Then as the**red**ball is replaced, there are still 4**red**balls left out of 9, so again there is a \dfrac{4}{9} chance of picking a**red**ball on the second selection.

**Step 2:**Use the tree diagram to determine the probability of selecting the same colour twice. We can see that there are two ways of doing this either blue and blue, or, red and red. We use the AND rule via the tree diagram, so

\textcolor{black}{\text{P(blue and blue)}=\dfrac{5}{9}\times\dfrac{5}{9}= \textcolor{blue}{\dfrac{25}{81}} \quad \text{ and } \quad \text{P(red and red)}=\dfrac{4}{9}\times\dfrac{4}{9}= \textcolor{red}{\dfrac{16}{81}} }

**Step 3:**The final step then is to add the probabilities together, by the OR rule for mutually exclusive events, to get,

\textcolor{black}{\text{P(same colour)}= \dfrac{25}{81} +\dfrac{16}{81}=\dfrac{41}{81}}

**Conditional Probability**

The **conditional probability** of A given B, is the “probability that event A happens **given that** event B happens”. You will not be told that it is a conditional probability question, but seeing words like ‘without replacement’ or ‘given’ will mean that it is one, or you may have to use your own intuition.

- If two events, A and B, are
**independent**, then

\textcolor{black}{\text{P}(A \text{ given } B) = \text{P}(B) \quad \text{ and } \quad \text{P}(B \text{ given } A) = \text{P}(A)}

- If two events, A and B are
**dependent**, then

\textcolor{black}{\text{P}(A \text{ and } B) = \text{P}(A) \times \text{P}(B \text{ given } A)}

**Example:**

Benjamin plays football for his local team. The probability that he is in the starting line up for his team this Sunday is 0.7. If he starts the game, the probability that he scores a goal is 0.4. **What is the probability that Benjamin starts the game but doesn’t score a goal?**

**Step 1:**We want to find \text{P(starts and doesn't score)}

Let “starts” be event A and “doesn’t score” be event B

**Step 2:**\text{P}(A) = 0.7

\text{P}(B \text{ given } A) = \text{P(doesn't score given he starts)} = 1 - 0.4 = 0.6

**Step 3:**Then,

\text{P}(A \text{ and } B) = \text{P}(A) \times \text{P}(B \text{ given } A) = 0.7 \times 0.6 = 0.42

**Conditional Probability Trees**

**Conditional probability trees** are similar to probability trees, but the probabilities change depending on the previous events.

**Example: **A bag contains 4 red balls and 5 blue balls. Raheem picks 2 balls at random. Calculate the probability that he selects the same coloured ball each time, given that each time a ball is selected, it is **not replaced**.

**Step 1:**Construct the probability tree showing two selections, noting that there are 9 balls to begin with, reducing to 8 after the first selection, as shown below, e.g. there is a \dfrac{4}{9} chance of picking a red ball, then since a red ball has been removed and not replaced, there are only 3 red balls left out of 8, so there is a \dfrac{3}{8} chance of picking a red ball on the second selection, and so on.

**Step 2:**Use the tree diagram to determine the probability of selecting the same colour twice. We can see that there are two ways of doing this either blue and blue, or, red and red. We use the AND rule via the tree diagram, so

\textcolor{black}{ \text{P(blue and blue)}=\dfrac{5}{9}\times\dfrac{4}{8}= \textcolor{blue}{\dfrac{20}{73}} \text{and} \text{P(red and red)}=\dfrac{4}{9}\times\dfrac{3}{8}= \textcolor{red}{\dfrac{12}{73}} }

**Step 3:**The final step then is to add the probabilities together, by the OR rule for mutually exclusive events, to get

\textcolor{black}{\text{P(Same colour)}= \dfrac{20}{73} +\dfrac{12}{73}=\dfrac{32}{73}}

### Example Questions

**Question 1: **Anna and Rob take their driving tests on the same day. The probability of Anna passing her driving test is 0.7. The probability of both Anna and Rob passing is 0.35

**(a)** Work out the probability of Rob passing his driving test.

**(b)** Work out the probability of both Anna and Rob failing their driving tests.

**[4 marks]**

**(a) **Let “Anna passing” be event A_p and “Rob” passing be event B_p.

To work out the probability of Rob passing we can write the probability of both passing as:

P(A_p \text{ and } R_p) =0.35

Substituting in the probability of Anna passing her test,

0.7 \times P(R_p) =0.35

Rearranging the equation to make P(R_p) the subject:

P(R_p) = 0.35 \div 0.7 = 0.5

**(b) ** The probability of both Anna and Rob failing their driving test can be found using a tree diagram as shown below:

Hence the probability of them both failing is \dfrac{3}{20} = 0.15.

**Question 2: **There are 12 counters in a bag, 7 are blue and the rest are green.

Sean takes out a counter from the bag a random then, without replacement, takes out another counter.

Work out the probability that the two counters Sean removes are the same colour.

**{3 marks]**

For this question when drawing the tree diagram we have to be careful as the probability changes between the two events. This is the result of not replacing the first counter hence only leaving 11 counters in the bag to pick from.

Adding together the probabilities of the result being blue then blue or green then green:

\dfrac{7}{22}+\dfrac{5}{33}=\dfrac{31}{66}

**Question 3: **The probability that a bus is on time is 0.75

Rory takes the bus to school two days a week. Calculate the probability the bus is late on each of those days.

**[3 marks]**

To work out the probability of the bus being late on both of the days we can use a tree diagram where E represents the bus being on time or early and L represents the bus being late.

Going bottom line we find that the probability of being late of both days is:

\dfrac{1}{16}

**Question 4: **** **There are 14 footballs in a bag, 9 have a blue pattern design and the rest have green pattern design. The coach takes out a ball out from the bag a random then, without replacement, takes out another one.

Is the coach more likely to pick out two balls that are the same colour or two that are different colours.

You **must** show your workings. Give your answer in its simplest form.

**[3 marks]**

Here we have to work out the probability that the coach takes out two balls that are a different colour.

For conditional probability questions, when drawing the tree diagram we have to be careful as the probability changes between the two events. This is the result of not replacing the first ball hence only leaving 13 balls in the bag to pick from.

Adding together the probabilities of the result being two different colours:

\dfrac{45}{182}+\dfrac{45}{182}=\dfrac{90}{182}=\dfrac{45}{91}

Hence as this is just below a half it is more likely that the coach picks two balls that are of the dame colour.

**Question 5: **William enters a badminton competition. The probability he wins a game is 0.6.

**(a)** Using this information complete the tree diagram shown below.

**(b) **Work out the probability that William wins at least one match.

**[4 marks]**

**(a)** The resultant tree diagram should look something like:

**(b) **To find the probability he wins at least one game we can simple add the top 3 branches probabilities together or subtract the probability of bottom branch from 1:

\dfrac{9}{25}+\dfrac{6}{25}+\dfrac{6}{25}=\dfrac{21}{25}

or,

1-\dfrac{4}{25}=\dfrac{21}{25}

### Worksheets and Exam Questions

#### (NEW) Probability and Tree Diagrams Exam Style Questions - MME

Level 4-5#### Probability - Drill Questions

Level 1-3#### Probability And Relative Frequency - Drill Questions

Level 4-5#### Tree Diagrams - Drill Questions

Level 4-5#### Probability Trees - Drill Questions

Level 6-7#### Probability and Tree Diagrams - Drill Questions

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