Conditional probability

In a local library, it is known that the probability that a member borrows a novel is $P(N) = 0,6$ and the probability that they borrow a comic is $P(C) = 0,4$. Furthermore, the probability that they borrow both is $P(N \cap C) = 0,15$.

Calculate the probability that a member borrows a comic if we know that they have already borrowed a novel.

A social media analyst studies the behaviour of teenagers on two apps: TikTok ($T$) and Instagram ($I$). The data indicate that:

• $70 \%$ of young people use TikTok.

• $50 \%$ use Instagram.

• The probability that a young person uses Instagram given that they use TikTok is $0,4$.

Calculate the probability that a randomly chosen young person uses both apps ($P(T \cap I)$) and reason whether the events are independent.

In a pencil case there are 5 fluorescent highlighters: 3 are yellow and 2 are pink. We draw two highlighters consecutively and without replacement (without putting the first one back in the pencil case).

Represent the situation with a tree diagram and calculate the probability that the second highlighter drawn is pink, given that the first one was yellow.

A Year 13 student has prepared 18 out of the 25 topics included in the final History exam. The exam consists of choosing two topics at random from the total and answering one of them.

1. a) Calculate the probability that the student knows both topics drawn.
2. b) Calculate the probability that the student passes the exam (that is, knows at least one of the two topics).

We have two opaque urns. Urn $U_A$ contains 4 white balls and 6 black balls. Urn $U_B$ contains 5 white balls and 3 black balls. The experiment consists of drawing a ball from urn $U_A$ and, without looking at its colour, placing it into urn $U_B$. Next, two balls are drawn simultaneously from urn $U_B$.

If the two balls drawn from the second urn turn out to be white, what is the probability that the ball transferred from the first urn was black? Express the result as an irreducible fraction.

In a Year 13 class, a survey was conducted on the use of glasses and the gender of students. The results are shown in the following contingency table:

If a student is chosen at random, calculate:

1. a) The probability that they are a boy, given that they wear glasses.
2. b) The probability that they wear glasses, given that they are a girl.

In a survey carried out on 200 students at a secondary school regarding social media use, it was observed that 120 use TikTok, 100 use Instagram and 60 use both platforms. If a student is chosen at random, calculate:

1. a) The probability that they use Instagram given that they already use TikTok.
2. b) The probability that they do not use TikTok if we know that they use Instagram.
3. c) Are the events "being a TikTok user" and "being an Instagram user" independent? Justify your answer mathematically.

A sports performance centre classifies its athletes according to their diet (Balanced or Deficient) and their performance in the last term (High or Low). The data are gathered in the following contingency table:

Calculate:

1. a) $P(R | D)$
2. b) $P(\bar{D} | \bar{R})$
3. c) Determine whether having a balanced diet and having a high performance are independent events.

Let $A$ and $B$ be two events of a random experiment such that the probability of $A$ is $P(A) = 0,5$, that of $B$ is $P(B) = 0,3$, and the probability of their union is $P(A \cup B) = 0,65$.

From the following Venn diagram, identify the shaded region and calculate $P(A|B)$.

Let $A$ and $B$ be two events of a random experiment such that $P(A) = 0,4$, $P(B) = 0,6$ and $P(A \cup B) = 0,8$.

1. a) Calculate $P(A | B)$.
2. b) Calculate $P(B | \bar{A})$.
3. c) Are $A$ and $B$ mutually exclusive events? Are they independent?

Let $A$ and $B$ be two events in a sample space such that we know $P(A \cup B) = 0,9$, $P(B|A) = 0,4$ and $P(A|\overline{B}) = 0,8$, where $\overline{B}$ represents the complementary event of $B$.

Using the properties of the algebra of events and the definition of conditional probability, calculate:

1. $P(A \cap B)$

2. $P(A)$

3. $P(B)$

Hint: Set up a system of equations using the definitions of conditional probability.