Basic Probability (AQA A-Level Mathematics): Revision Notes

Example: Tossing a Coin Twice Let's say you toss a coin twice. The possible outcomes are:

1. First Toss: Heads ($H$) or Tails ($T$)
2. Second Toss: Heads ($H$) or Tails ($T$) The tree diagram would look like this:

          Start
           /  \
          H    T   (First Toss)
         / \  / \
        H   T H   T (Second Toss)

Example:

• Probability of getting Heads followed by Heads (HH):
• $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
• Probability of getting Heads followed by Tails (HT):
• $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
• Probability of getting Tails followed by Heads (TH):
• $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
• Probability of getting Tails followed by Tails (TT):
• $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Example: Drawing Coloured Balls Imagine a bag contains 2 red balls and 1 blue ball.

You draw a ball, record its colour, and then draw another ball without replacing the first.

3. First Draw:

• Red (R) or Blue (B)

4. Second Draw (if first draw was Red):

• Red (1 remaining) or Blue

5. Second Draw (if first draw was Blue):

• Red (2 remaining) The tree diagram would show the probabilities of each sequence of draws, helping you calculate probabilities such as the likelihood of drawing two red balls in a row.

Example:

• Probability of drawing two red balls:
• First Red: $\frac{2}{3}$
• Second Red: $\frac{1}{2}$
• Total Probability: $\frac{2}{3} \times \frac{1}{2} = \frac{1}{3}$

Example Questions

A bag contains five red and four blue tokens. A token is chosen at random, and the colour is recorded. The token is not replaced. A second token is chosen, and the colour is recorded.

Tasks:

6. Draw a tree diagram to illustrate this situation. Find the probability that:

7. The second token is red given that the first token is blue.

8. The first token is red given that the second token is blue.

9. The first token is blue given that the tokens are different colours.

10. The tokens are the same colour given that the second token is red.

Worked Solutions

a) Draw a tree diagram to illustrate this situation:

The tree diagram is drawn with the first level representing the first token draw, which can be red ($R$) or blue ($B$).

The probabilities are:

The second level represents the second token draw, contingent on the first draw's result.

If the first token was red:

If the first token was blue:

b) Find the probability that the second token is red given that the first token is blue.

The probability that needs to be calculated can be represented as this:

Given that the first token is blue, we ignore the first branch leading to red and focus on the scenario where the first token was blue.

We then calculate the probability of drawing a red token next:

c) Find the probability that the first token is red given that the second token is blue.

The probability that needs to be calculated can be represented as this:

Calculate the probability of the first token being red and the second token being blue:

Calculate the probability of the first token being blue and the second token being blue:

The total probability of the desired outcome is:

Which simplifies to

d) Find the probability that the first token is blue given that the tokens are different colours.

The probability that needs to be calculated can be represented as this:

Calculate the probability of the first token being blue and the second being red:

The total probability for different colours is:

Which is further simplified to

e) Find the probability that the tokens are the same colour given that the second token is red.

The probability that needs to be calculated can be represented as this:

Given that the second token is red:

The probability that the first token was red is calculated as:

The probability that the first token was blue is:

The total probability of the second token being red regardless of the first token is the sum of these probabilities:

To find the probability that the two tokens are the same colour, given that the second token is red, use: