Advanced level 2025 BAEBOC Pure mathematics with statistics 3

Advanced level 2025 BAEBOC Pure mathematics with statistics 3

Advanced level 2025 BAEBOC Pure mathematics with statistics 3

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1. i) Events X and Y are such that P(X)=31​, P(Y)=41​ and P(X∪Y)=21​. Find a) $P(X\cap Y)$ b) $P(Y|X)$ c) P(X′∩Y′)

ii) In a restaurant 40% of the customers choose cake for their main meal. If a customer chooses cake, the probability that he will choose ice cream to follow is 0.6. If he does not have cake, the probability that he will choose ice cream is 0.3. Find the probability that a customers picked at random will choose a) ice Cream b) Cake given that he had chosen ice cream.

2. The table above shows the yield, in litres, of milk produced by 131 cows on a certain farm on a given day.

Calculate a) The median yield b) Draw a cumulative frequency curve and from it estimate the semi-interquartile range c) Calculate the mean and standard deviation of the distribution

3. The discrete random variable X can take only the values 0, 1, 2, 3, 4, 5. The probability distribution of X is given by the following ¹ $P(X=0)=3k$, $P(X=1)=3k$, $P(X=2)=k$, $P(X=3)=k$, $P(X=4)=2k$, $P(X=5)=b$ $P(X\ge 2)=3P(X<2)$ Where a and b are constants a) Determine the values of a and b b) Find the median of X c) Show that the expectation of X is 825​ and determine the variance of X.  

4. The mean number of flaws per 100m of material produced on a certain machine at Mankon Fabrics is 2. If the flaws occur randomly, find the probability that a) in a 100m length of material there will be no flaws b) in a 200m length of material there will be more than 3 flaws c) Many materials of length 100m are taken at random and checked. What is the most common number of flaws per material to be noticed. d) If there is a 10% probability of having at least one flaws on a 100m length material produced by another machine at Mankon Fabrics. Find the mean number flaws per 100m of material produced by this machine.

5. The continuous random variable X has probability density function given by