A company manufactures chocolate tablets of 100 grams. The quality control department performs several types of control.

Part A Control before market release

A chocolate tablet must weigh 100 grams with a tolerance of two grams more or less. It is therefore put on the market if its mass is between 98 and 102 grams. The mass (expressed in grams) of a chocolate tablet can be modelled by a random variable $X$ following the normal distribution with mean $\mu = 100$ and standard deviation $\sigma = 1$. The adjustment of the manufacturing chain machines allows us to modify the value of $\sigma$.

1. Calculate the probability of the event $M$ : ``the tablet is put on the market''.
2. We wish to modify the adjustment of the machines so that the probability of this event reaches 0.97. Determine the value of $\sigma$ so that the probability of the event ``the tablet is put on the market'' equals 0.97.

Part B Control upon reception

The department controls the quality of cocoa beans delivered by producers. One of the quality criteria is the moisture content which must be $7\%$. The bean is then said to be compliant. The company has three different suppliers: the first supplier provides half of the bean stock, the second $30\%$ and the last provides $20\%$ of the stock. For the first, $98\%$ of its production respects the moisture content; for the second, which is somewhat cheaper, $90\%$ of its production is compliant, and the third supplies $20\%$ of non-compliant beans. We randomly choose a bean from the received stock. We denote $F _ { i }$ the event ``the bean comes from supplier $i$'', for $i$ taking the values 1, 2 or 3, and $C$ the event ``the bean is compliant''.

1. Determine the probability that the bean comes from supplier 1, given that it is compliant.

Consider $T$ the random variable following the normal distribution with mean $\mu = 60$ and standard deviation $\sigma = 6$.
The probability $P _ { ( T > 60 ) } ( T > 72 )$ rounded to the nearest thousandth is: Answer A: 0.954 Answer B: 1 Answer C: 0.023 Answer D: 0.046

Out of the 14 students taking a test, 5 are well prepared, 6 are adequately prepared and 3 are poorly prepared. There are 10 questions on the test paper. A well prepared student can answer 9 questions correctly, an adequately prepared student can answer 6 questions correctly and a poorly prepared student can answer only 3 questions correctly.
For each probability below, write your final answer as a rational number in lowest form.
(a) If a randomly chosen student is asked two distinct randomly chosen questions from the test, what is the probability that the student will answer both questions correctly?
Note: The student and the questions are chosen independently of each other. "Random" means that each individual student/each pair of questions is equally likely to be chosen.
(b) Now suppose that a student was chosen at random and asked two randomly chosen questions from the exam, and moreover did answer both questions correctly. Find the probability that the chosen student was well prepared.