Advanced level 2025 South West Regional Mock mathematics statistics 3

Advanced level 2025 South West Regional Mock mathematics statistics 3

Advanced level 2025 South West Regional Mock mathematics statistics 3

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1. (i) Two events A and B are such that P(A)=32​, P(B)=21​ and P(A∪B)=65​. Find: a) $P(A\cap B)$ b) P(B∣A′)

State with reason(s) whether or not A and B are: c) Independent d) Mutually exclusive

(ii) In a certain country 70% of the defendants being tried in the law Courts actually committed the crime. For those who committed the crime, the probability of being found guilty is 0.8. ¹ For those who did not commit the crime, the probability of being found guilty is 0.05. a) Draw a tree diagram to show all the possible outcomes. b) Find the probability that a randomly chosen defendant is found guilty.  

2. The marks scored by students in a Biology examination is summarized in the table below:

Calculate to 2 decimal places, the a) Mean of the distribution. b) Standard deviation of the distribution. c) Mode of the distribution. d) Median of the distribution.

3. i. A student is waiting for a bus. Past experience shows that 4% of vehicles in the City are buses. Find the: a) Mean number of vehicles up to and including the first bus. b) Probability to 4 decimal places that the first bus is the 6th vehicle to arrive. c) State the condition (s) for Poisson distribution to approximate the Binomial distribution.

ii. The probability that a mango is bad is 0.5%. Mangoes are packed in baskets of 600. Calculate to 4 decimal places, the probability that in a randomly selected basket, there are a) Two bad mangoes b) More than three bad mangoes.

4. The probability mass function $f$, of a discrete random variable $x$, is defined by f(x)=⎩” />⎨” />⎧​cx−1​c9−x​0​for x=2,3,4for x=6,7,8Otherwise​ Calculate: a) The value of the constant $c$. b) The mean and variance of $x$. c) $E(4x-2)$ and $Var(4x-2)$.