Practice Questions

If E and F are events of a sample space S such that $P(E'|F) = 0.6$, calculate $P(E|F)$.

Given $P(A|B) = P(A)$, where $P(A) > 0$ and $P(B) > 0$. Evaluate the relationship between $P(B|A)$ and $P(B)$ and justify your conclusion.

State the mathematical condition that must be satisfied for two events, A and B, to be considered independent.

Recall and state the relationship between the conditional probabilities $P(E'|F)$ and $P(E|F)$.

List the three conditions that a set of events $E_1, E_2, ..., E_n$ must satisfy to be considered a partition of a sample space S.

Describe in words what the notation $P(A|B)$ represents in the context of probability. Also, write down the formula used to calculate it.

Events A and B are independent. If $P(A) = \frac{1}{3}$ and $P(B) = \frac{3}{4}$, calculate $P(A \cap B)$.

A single card is drawn from a standard deck of 52 playing cards. If it is known that the card is a red card, calculate the probability that it is a king.

A speaks the truth in 75% of cases and B in 80% of cases. Formulate an expression for the probability that they contradict each other in stating the same fact. Assume their statements are independent, and evaluate this probability.

Define the conditional probability of an event E, given that event F has already occurred. Assume $P(F) \neq 0$.

What does it mean for a set of events $E_1, E_2, ..., E_n$ to be exhaustive with respect to a sample space S?

Given that E and F are events such that $P(E) = 0.5$, $P(F) = 0.4$ and $P(E \cap F) = 0.2$, calculate $P(E|F)$.

A and B are independent events such that $P(A) = 0.4$ and $P(B) = 0.5$. Calculate $P(A' \cap B')$.

Two fair dice are rolled. It is known that the sum of the numbers on the dice is exactly 8. Calculate the probability that one of the dice shows the number 3.

Recall the definition of independent events. Use this definition to identify if events A and B are independent, given that $P(A) = \frac{1}{4}$, $P(B) = \frac{1}{3}$, and $P(A \cap B) = \frac{1}{12}$.

A student claims that if two events A and B are mutually exclusive, they must also be independent. Justify whether this statement is correct or incorrect for non-empty events A and B.

State the multiplication rule of probability for two events E and F associated with a sample space S.

Explain the Theorem of Total Probability and state the formula associated with it for an event A and a partition ${E_1, E_2, ..., E_n}$ of the sample space S.

If A and B are two events of a sample space S, and F is an event of S such that $P(F) \neq 0$, state the formula for $P((A \cup B)|F)$. How does this formula simplify if A and B are disjoint events?

State Property 1 of conditional probability, which gives the values of $P(S|F)$ and $P(F|F)$. Explain mathematically why $P(S|F) = 1$.

Given $P(A) = 0.6$, $P(B) = 0.5$ and $P(A \cap B) = 0.25$. Analyze if events A and B are independent.

A bag contains 6 white and 4 black balls. Two balls are drawn at random one after the other without replacement. Calculate the probability that the first ball is white and the second is black.

If $P(A) = \frac{2}{5}$, $P(B) = \frac{1}{3}$ and $P(A|B) = \frac{1}{2}$, calculate $P(A \cup B)$.

A factory has three machines X, Y, and Z, producing 1000, 2000, and 3000 bolts per day, respectively. Machine X produces 1% defective bolts, Y produces 1.5% defective bolts, and Z produces 2% defective bolts. A bolt is chosen at random at the end of the day. Calculate the probability that it is defective.

A man is known to speak the truth 3 out of 5 times. He throws a die and reports that the number obtained is a 4. Calculate the probability that the number obtained is actually a 4.

Formulate a simple experiment using a single fair die and define two events, E and F, such that $P(E|F) = 1$ but $E \neq F$.

Prove that if E and F are two independent events, then their complements, $E'$ and $F'$, are also independent.

Starting from the definition of conditional probability, derive the general multiplication rule for three events: $P(E \cap F \cap G) = P(E) P(F|E) P(G|E \cap F)$.

Create a realistic problem involving a medical test's accuracy that requires the use of Bayes' theorem. The problem must define the disease prevalence, test sensitivity (true positive rate), and specificity (true negative rate). Then, solve the problem you created.

A student solves the following problem: "A bag contains 3 red and 5 black balls. Two balls are drawn successively without replacement. What is the probability that the second ball is red?" Student's solution: $P(\text{2nd is Red}) = P(\text{1st is Red AND 2nd is Red}) + P(\text{1st is Black AND 2nd is Red}) = (3/8) \times (2/7) + (5/8) \times (3/7) = 6/56 + 15/56 = 21/56 = 3/8$. The student then claims that since $P(\text{1st is Red}) = 3/8$, the events "first ball is red" and "second ball is red" must be independent. Critique the student's reasoning about independence and justify mathematically whether the events are independent.

Two players, A and B, take turns shooting at a target. A has a probability of $1/3$ of hitting, and B has a probability of $1/4$. They continue until one hits. If A starts, formulate the probability of A winning and evaluate it.

State Bayes' Theorem for a partition of sample space ${E_1, E_2, ..., E_n}$ and an event A. Explain the meaning of the terms 'priori probability' and 'posteriori probability' in the context of this theorem.

Describe the multiplication rule of probability for two events, and then state its extension for three events E, F, and G. Explain in words what each term in the three-event formula, $P(E \cap F \cap G) = P(E) \cdot P(F|E) \cdot P(G|E \cap F)$, represents.

There are two urns. Urn I contains 3 red and 5 black balls. Urn II contains 4 red and 2 black balls. One urn is chosen at random and a ball is drawn. The ball is found to be red. Calculate the probability that it was drawn from Urn I.

Explain the fundamental difference between mutually exclusive events and independent events. Can two events with non-zero probabilities be both mutually exclusive and independent? Explain your reasoning.

Consider an experiment of tossing a fair coin until a head appears for the first time. Let E be the event that the number of tosses required is even, and F be the event that the number of tosses required is greater than 2. Evaluate whether E and F are independent events.

Justify why for any two events E and F from a sample space S, if $E \subset F$, then $P(E|F) \geq P(E)$.

Summarize the concept of conditional probability using the experiment of rolling a single fair die. Define event E as 'getting an even number' and event F as 'getting a number greater than 3'. Describe how knowing that F has occurred changes the probability of E occurring.

A diagnostic test for a disease has 98% accuracy in detecting the disease if a person has it, and 99% accuracy in correctly identifying that a person does not have the disease if they are healthy. It is known that 1% of the population has the disease. If a person is selected at random and tests positive, calculate the probability that they actually have the disease.

A researcher claims that for any partition of a sample space $\{E_1, E_2, ..., E_n\}$ and any event $A$ with $P(A) > 0$, the sum $\sum_{i=1}^{n} P(E_i|A)$ must equal 1. Evaluate this claim and provide a justification.

A company has three machines (A, B, C) producing bolts. Machine A produces 50%, B produces 30%, and C produces 20%. The defect rates are 2%, 3%, and 5% respectively. A bolt is selected and found to be defective. A manager proposes that it is most likely from machine C because it has the highest defect rate. Justify, using Bayes' Theorem, whether the manager's proposal is mathematically sound.

Let $\{E_1, E_2, E_3\}$ be a partition of the sample space S. Let A be an event with $P(A) > 0$. Prove that if $P(A|E_1) = P(A|E_2) = P(A|E_3)$, then $P(E_i|A) = P(E_i)$ for $i = 1, 2, 3$. Justify what this result implies about the relationship between event A and the events in the partition.

Three fair coins are tossed. Let E be the event 'at most one head appears' and F be the event 'both heads and tails appear'. Examine if E and F are independent events.

Box A contains 1 white, 2 red and 3 black balls. Box B contains 2 white, 3 red and 1 black ball. One ball is transferred from Box A to Box B, and then a ball is drawn from Box B. The ball drawn is found to be red. Calculate the probability that the transferred ball was black.

Design a game for two players, P1 and P2, using a single biased coin where $P(\text{Head}) = p$. The game ends when the first Head appears. P1 wins if the first Head appears on an odd-numbered toss (1st, 3rd, 5th, ...), and P2 wins if it appears on an even-numbered toss (2nd, 4th, 6th, ...). Formulate and evaluate their respective probabilities of winning in terms of $p$.