4MA0 Higher June 2016 Paper 4 Q16
16 In a bag there is a total of 20 coins.
10 coins are 20 cent coins
6 coins are 10 cent coins
4 coins are 5 cent coins
Emma takes at random two of the coins from the bag.
(a) Complete the probability tree diagram.
(b) Work out the probability that Emma takes two 5 cent coins. (2)
(c) Work out the probability that the total value of the two coins is 20 cents or less. (3)
| Scheme | Marks |
|---|---|
| \(\dfrac{6}{20}, \dfrac{4}{20}\) | B1 |
| \(\dfrac{9}{19}\;\; \dfrac{6}{19}\;\; \dfrac{4}{19}\;\; \dfrac{10}{19}\;\; \dfrac{5}{19}\;\; \dfrac{4}{19}\;\; \dfrac{10}{19}\;\; \dfrac{6}{19}\;\; \dfrac{3}{19}\) | B1 |
| (2) |
Notes
B1: For \(\dfrac{6}{20}, \dfrac{4}{20}\) correct on LH branches
B1: For all other branches correct
| Scheme | Marks |
|---|---|
| \(\dfrac{4}{20} \times \dfrac{3}{19}\) | M1ft |
| \(\dfrac{12}{380}\) oe | A1ft |
| (2) |
Notes
M1ft: From their Tree diagram
A1ft: From their Tree diagram
oe. Eg \(\dfrac{3}{95}\)
Accept 0.031(57…) rounded or truncated to at least 3 decimal places.
| Scheme | Marks |
|---|---|
\(\dfrac{6}{20} \times \dfrac{5}{19}\) or 0.078(947 … ) or \(\dfrac{6}{20} \times \dfrac{4}{19}\) or 0.063(157 … ) or \(\dfrac{4}{20} \times \dfrac{3}{19}\) or 0.031(578…) | M1ft |
| \(\dfrac{6}{20} \times \dfrac{5}{19} + \dfrac{6}{20} \times \dfrac{4}{19} + \dfrac{4}{20} \times \dfrac{6}{19} + \dfrac{4}{20} \times \dfrac{3}{19}\) | M1ft |
| \(\dfrac{90}{380}\) oe | A1 |
| (3) |
Notes
M1ft: For one correct product from their Tree diagram
M1ft: For sum of all correct products from their Tree diagram
A1: For \(\dfrac{9}{38}\) oe or 0.236(842…)
NB: Accept use of decimals if rounded or truncated to at least 3 decimal places.
With Replacement
| Scheme | Marks |
|---|---|
| \(\dfrac{6}{20} \times \dfrac{6}{20}\) or 0.09 or \(\dfrac{6}{20} \times \dfrac{4}{20}\) or 0.06 or \(\dfrac{4}{20} \times \dfrac{4}{20}\) or 0.04 | M1 |
| \(\dfrac{6}{20} \times \dfrac{6}{20} + \dfrac{6}{20} \times \dfrac{4}{20} + \dfrac{4}{20} \times \dfrac{6}{20} + \dfrac{4}{20} \times \dfrac{4}{20}\) or \(\dfrac{100}{400}\) or 0.25 oe | M1 |
Alternative method
| Scheme | Marks |
|---|---|
Eg \(1 - \left(\dfrac{10}{20} \times \dfrac{9}{19} + \dfrac{10}{20} \times \dfrac{6}{19} + \dfrac{10}{20} \times \dfrac{4}{19} + \dfrac{6}{20} \times \dfrac{10}{19} + \dfrac{4}{20} \times \dfrac{10}{19}\right)\) or \(\dfrac{10}{20} \times \dfrac{9}{19}\) oe | M2 |
| \(\dfrac{90}{380}\) oe | A1 |
Notes
M2: For a complete method.
Ft from their Tree diagram
A1: For \(\dfrac{9}{38}\) oe or 0.236(842…)
NB: Accept use of decimals if rounded or truncated to at least 3 decimal places.