Conditional Probability and Multiplication Theorem

If $A\text{ and }B$ are two events, such that $P\left(A\right)=\frac{3}{8},P\left(B\right)=\frac{5}{8}\text{ and }P(A\cup B)=\frac{3}{4}$, then find $P\left(\frac{A}{B}\right)$.

Jake and Elisa are given a mathematics problem. The probability that Jake can solve it is $0.35$. If Jake has solved it, the probability that Elisa can solve it is $0.6$. Otherwise, the probability that Elisa can solve it is $0.45$. Draw a tree diagram to illustrate the above question, showing clearly the probabilities on each branch. Find the probability that at least one of the students can solve the problem. 

Jake and Elisa are given a mathematics problem. The probability that Jake can solve it is $0.35$. If Jake has solved it, the probability that Elisa can solve it is $0.6$. Otherwise, the probability that Elisa can solve it is $0.45$. Draw a tree diagram to illustrate the above question, showing clearly the probabilities on each branch. Find the probability that Jake solves the problem, given than Elisa has.

Jake and Elisa are given a mathematics problem. The probability that Jake can solve it is $0.35$. If Jake has solved it, the probability that Elisa can solve it is $0.6$. Otherwise, the probability that Elisa can solve it is $0.45$. Draw a tree diagram to illustrate the above question, showing clearly the probabilities on each branch.

The Venn diagram illustrates the number of students taking each of the three sciences: physics, chemistry and biology.

A student is randomly chosen from the group.

Find the probability that the student studies physics, given that they study chemistry. (Write answer in simplest fraction form)

$Q~N\left(4.03, 0.7^2\right)$. Find the probability $P\left(Q<4.9|Q>2.9\right)$ with technology.

Two fair tetrahedral dice with faces numbered $1, 2, 3$ and $4$ are thrown. The discrete random variable $D$ is defined as the product of the two numbers thrown. Find $P\left(D \mathrm{is} \mathrm{a} \mathrm{square} \mathrm{number} | D<8\right)$.

A group of $50$ investors own properties in north European cities. The following Venn diagram shows how many investors own properties in Amsterdam, Brussels or Cologne. One of the investors is chosen at random.

You are given that $P\left(C\right|B)=\frac{10}{23}$ and $P\left(A\right|C)=\frac{1}{3}$. Calculate the remaining regions shown in the Venn Diagram.

A supermarket uses two suppliers, $C$ and $D$ of strawberries. Supplier $C$ supplies $70\%$ of the supermarkets's strawberries. Strawberries are examined in a quality control inspection ($QCI$): $90\%$ of the strawberries supplied by $C$ pass $QCI$ and $95\%$ of the strawberries from $D$ pass $QCI$.

A strawberry is selected at random.

Given that the strawberry passes $QCI$, find the probability that it came from supplier $D$.

(Round off and Write answer up to $3$ decimal places)

A group of $50$ investors own properties in north European cities. The following Venn diagram shows how many investors own properties in Amsterdam, Brussels or Cologne. One of the investors is chosen at random.

Find $P\left(C\right|A).$(Round off and write up to $2$ decimal places )

A group of $50$ investors own properties in north European cities. The following Venn diagram shows how many investors own properties in Amsterdam, Brussels or Cologne. One of the investors is chosen at random.

Find $P\left(B\right|A).$ (Round off and write up to $2$ decimal places)

The probability distribution of a discrete random variable $A$ is defined by this table:

Find $\mathrm{P}\left(A>8|A\le 11\right)$.

(Write answer in fraction form).

If $P\left(A\right)=\frac{7}{13},P\left(B\right)=\frac{9}{13}\text{ and }P(A\cap B)=\frac{4}{13},$ find $P(A/B)$.

If $P\left(E\right)=0.6, P\left(F\right)=0.3$ and $P(E\cap F)=0.2$, then find $P\left(F|E\right)$.

If $P\left(A\right)=0.6, P\left(B\right)=0.3$ and $P(A\cap B)=0.2$ then find $P\left(\frac{A}{B}\right)$

The letters of the word "$\mathrm{LOGARITHM}$" are arranged at random. The probability that arrangements starts with vowel and end with consonant is

A man is known to speak the truth $2$ out of $3$ times. If he throws a die and reports that it is six, then the probability that it is actually five is

When two dice are rolled, let $x$ be the probability of getting a sum of the numbers appear on the dice is at most $7$. Let $y$ be the probability of getting a sum $7$ at least once when a pair of dice are rolled $n$ times. In order to have $y>x,$ the minimum $n$ is

A coin is tossed three times, where event $E$: head on third toss, event $F$: heads on first two tosses. Determine $P\left(E|F\right)$.