1

Nov 2010 p61 q5

Three friends, Rick, Brenda and Ali, go to a football match but forget to say which entrance to the ground they will meet at. There are four entrances, \( A, B, C \) and \( D \). Each friend chooses an entrance independently.
- The probability that Rick chooses entrance \( A \) is \( \frac{1}{3} \). The probabilities that he chooses entrances \( B, C \) or \( D \) are all equal.
- Brenda is equally likely to choose any of the four entrances.
- The probability that Ali chooses entrance \( C \) is \( \frac{2}{7} \) and the probability that he chooses entrance \( D \) is \( \frac{3}{5} \). The probabilities that he chooses the other two entrances are equal.
i Find the probability that at least 2 friends will choose entrance \( B \).
ii Find the probability that the three friends will all choose the same entrance.

\[ \begin{aligned} \mathrm{P}(\text { at least two choose } B)= & \mathrm{P}(\text { two choose } B)+\mathrm{P}(\text { three choose } B) \\ = & \mathrm{P}\left(\mathrm{abr}^{\prime}\right)+\mathrm{P}\left(\mathrm{ab}^{\prime} \mathrm{r}\right)+\mathrm{P}\left(\mathrm{a}^{\prime} \mathrm{br}\right)+\mathrm{P}(\mathrm{abr}) \\ = & \left(\frac{2}{35} \times \frac{1}{4} \times \frac{7}{9}\right)+\left(\frac{2}{35} \times \frac{3}{4} \times \frac{2}{9}\right)+\left(\frac{33}{35} \times \frac{1}{4} \times \frac{2}{9}\right) \\ & +\left(\frac{2}{35} \times \frac{1}{4} \times \frac{2}{9}\right) \\ = & \frac{8}{105} \text { or } 0.0762 \end{aligned} \] ii \( \quad \mathrm{P}( \) all \( A)+\mathrm{P}( \) all \( B)+\mathrm{P}( \) all \( C)+\mathrm{P}( \) all \( D) \) \[ \left(\frac{2}{35} \times \frac{1}{4} \times \frac{3}{9}\right)+\left(\frac{2}{35} \times \frac{1}{4} \times \frac{2}{9}\right)+\left(\frac{10}{35} \times \frac{1}{4} \times \frac{2}{9}\right)+\left(\frac{21}{35} \times \frac{1}{4} \times \frac{2}{9}\right)=\frac{2}{35} \text { or } 0.571 \]

2

Nov 2023 p53 q3a

Tim has two bags of marbles, \( A \) and \( B \).
Bag \( A \) contains 8 white, 4 red and 3 yellow marbles,
Bag \( B \) contains 6 white, 7 red and 2 yellow marbles.
Tim also has an ordinary fair 6 -sided dice. He rolls the dice. If he obtains a 1 or 2 , he chooses two marbles at random from bag \( A \), without replacement. If he obtains a \( 3,4,5 \) or 6 , he chooses two marbles at random from bag \( A \), without replacement marbles at random from bag \( B \), without replacement.
Find the probability that both marbles are white.

3

June 2023 p52 q2

A sports event is taking place for 4 days, beginning on Sunday. The probability that it will rain on Sunday is 0.4 . On any subsequent day, the probability that it will rain is 0.7 if it rained on the previous day and 0.2 if it did not rain on the previous day.
(a) Find the probability that it does not rain on any of the 4 days of the event.
(b) Find the probability that the first day on which it rains during the event is Tuesday.
(c) Find the probability that it rains on exactly one of the 4 days of the event.

4

MARCH 2023 P52 Q5

Marco has four boxes labelled K, L, M and N. He places them in a straight line in the order K, L, M, N with K on the left. Marco also has four coloured marbles: one is red, one is green, one is white and one is yellow. He places a single marble in each box, at random. Events A and B are defined as follows. A: The white marble is in either box L or box M. B: The red marble is to the left of both the green marble and the yellow marble. Determine whether or not events A and B are independent.