IB Maths AA HL 4.2 Statistics and Probability Ahl Content Questions

Amanda enters data from surveys into a database. It can be assumed that the accuracy of any survey entered is independent of all other surveys entered.
From previous records, it is known that Amanda enters 8 % of the surveys inaccurately.

Find the probability that a randomly selected survey was

entered by Amanda, given that the survey was entered inaccurately.

The following year, the accuracy of Amanda's and Bryce's work remained the same, as did the percentage of surveys entered by each of the three employees. However, Carmen's accuracy had improved and the probability that she entered a survey inaccurately was now x %.

The probability that a randomly selected survey had been entered inaccurately was now the same as the probability that Carmen made an error when entering a survey.

The random variable X has probability distribution $\operatorname{Po}(8)$.

5.
Two species of plant, A and B , are identical in appearance though it is known that the mean length of leaves from a plant of species A is 5.2 cm , whereas the mean length of leaves from a plant of species B is 4.6 cm . Both lengths can be modelled by normal distributions with standard deviation 1.2 cm .
In order to test whether a particular plant is from species A or species B, 16 leaves are collected at random from the plant. The length, x, of each leaf is measured and the mean length evaluated. A one-tailed test of the sample mean, $\bar{X}$, is then performed at the 5 % level, with the hypotheses: $H_{0}: \mu=5.2$ and $H_{1}: \mu<5.2$.

It is now known that in the area in which the plant was found $90 \%$ of all the plants are of species A and $10 \%$ are of species B.

If, having done the test, the sample mean is found to lie within the critical region, find the probability that the leaves came from a plant of species A .