Math / Data & Statistics

QuestionThe table below shows the number of coins in a purse. Two of the coins are picked at random and taken out of the purse at the same time. a) If one of the coins picked is a 20 p piece, is it possible for the total of the two coins to be exactly 60 p? b) Work out the probability that the total of the two coins is exactly 60p. Give your answer as a fraction in its simplest form. \begin{tabular}{|c|c|c|c|c|c|} \hline Coin & $5 p$ & $10 p$ & $20 p$ & $50 p$ & $£ 1$ \\ \hline Frequency & 1 & 3 & 2 & 5 & 1 \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. The purse contains coins with the following frequencies: $5p: 1$ , $10p: 3$ , $20p: 2$ , $50p: 5$ , $£1: 1$ .
2. We are selecting 2 coins at random from the purse.
3. We need to determine if it's possible for the total of two coins to be exactly $60p$ .
4. We need to calculate the probability of the total being $60p$ .

STEP 2

1. Determine if a combination with a 20p coin can sum to 60p.
2. Calculate the total number of ways to pick 2 coins.
3. Identify combinations that sum to 60p.
4. Calculate the probability of selecting a combination that sums to 60p.

STEP 3

List possible combinations with a 20p coin:
- $20p + 40p = 60p$ (not possible as there is no 40p coin) - $20p + 20p = 40p$ (not 60p) - $20p + 50p = 70p$ (not 60p) - $20p + 10p = 30p$ (not 60p) - $20p + 5p = 25p$ (not 60p) - $20p + £1 = 120p$ (not 60p)
Conclusion: It is not possible for the total to be exactly 60p if one coin is a 20p piece.

STEP 4

Calculate the total number of ways to pick 2 coins from the purse:
Total coins = $1 + 3 + 2 + 5 + 1 = 12$
Number of ways to pick 2 coins from 12:
$\binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 66$

STEP 5

Identify combinations that sum to 60p:
- $10p + 50p = 60p$
Frequency of $10p$ coins = 3 Frequency of $50p$ coins = 5
Number of ways to pick one $10p$ and one $50p$ :
$3 \times 5 = 15$

STEP 6

Calculate the probability of selecting a combination that sums to 60p:
Probability = $\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$
$\frac{15}{66} = \frac{5}{22}$
The probability that the total of the two coins is exactly 60p is:
$\frac{5}{22}$

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