## Exercise 1

Let $X$ be a discrete random variable with the following probability distribution

- Calculate and represent graphically the distribution function.
- Calculate the following probabilities a. $P(X<7.5)$. b. $P(X>8)$. c. $P(4\leq X\leq 6.5)$. d. $P(5<X<6)$.

## Exercise 2

Let $X$ be a discrete random variable with the following probability distribution

- Calculate the probability function.
- Calculate the following probabilities a. $P(X=6)$. b. $P(X=5)$. c. $P(2<X<5.5)$. d. $P(0\leq X<4)$.
- Calculate the mean.
- Calculate the standard deviation.

## Exercise 3

An experiment consist in injecting a virus to three rats and checking if they survive or not. It is known that the probability of surviving is $0.5$ for the first rat, $0.4$ for the second and $0.3$ for the third.

- Calculate the probability function of the variable $X$ that measures the number of surviving rats.
- Calculate the distribution function.
- Calculate $P(X\leq 1)$, $P(X\geq 2)$ and $P(X=1.5)$.
- Calculate the mean and the standard deviation. Is representative the mean?

## Exercise 4

The chance of being cured with certain treatment is 0.85. If we apply the treatment to 6 patients,

- What is the probability that half of them get cured?
- What is the probability that a least 4 of them get cured?

## Exercise 5

Ten persons came into contact with a person infected with tuberculosis. The probability of being infected after contacting a person with tuberculosis is 0.1.

- What is the probability that nobody is infected?
- What is the probability that at least 2 persons are infected?
- What is the expected number of infected persons?

## Exercise 6

The probability of suffering an adverse reaction to a vaccine is 0.001. If 2000 persons are vaccinated, what is the probability of suffering some adverse reaction?

## Exercise 7

The average number of calls per minute received by a telephone switchboard is 120.

- What is the probability of receiving less than 4 calls in 2 seconds?
- What is the probability of receiving at least 3 calls in 3 seconds?

## Exercise 8

A test contains 10 questions with 3 possible options each. For every question you get a point if you give the right answer and lose half a point if the answer is wrong. A student knows the right answer for 3 of the 10 questions and answers the rest randomly. What is the probability of passing the exam?

## Exercise 9

It has been observed experimentally that 1 of every 20 trillions of cells exposed to radiation mutates becoming carcinogenic. We know that the human body has approximately 1 trillion of cells by kilogram ot tissue. Calculate the probability that a 60 kg person exposed to radiation develops cancer. If the radiation affects 3 persons weighing 60 kg, what is the probability that a least one of them develops cancer?

## Exercise 10

A diagnostic test for a disease returns 1% of positive outcomes, and the positivie and negative predictive values are 0.95 and 0.98 respectively.

- Calculate the prevalence of the disease.
- Calculate the sensitivity and the specificity of the test.
- If the test is applied to 12 sick persons, what is the probability of getting at least a wrong diagnosis?
- If the test is applied to 12 persons, what is the probability of getting a right diagnosis for all of them?

## Exercise 11

In a study about a parasite that attacks the kidney of rats it is known that the average number of parasites per kidney is 3.

- Calculate the probability that a rat has more than 3 parasites.
- Calculate the probability of having at least 9 rats infected in a sample of 10 rats.

## Exercise 12

In a physiotherapy course there are 60% of females and 40% of males.

- If 6 random students have to go to a hospital for making practices, what is the probability of going more males than females?
- In 5 samples of 6 students, what is the probability of having some sample without males?