225 practice questions covering Probability, organised into 4 topics.
225 questions include a written explanation.
Pick a topic below, or try the samples first.
A bag holds 6 white balls, 5 black balls and 4 yellow balls, alike apart from colour. One ball is drawn at random. What is the probability that the ball drawn is white or yellow?
A$\frac{11}{15}$
B$\frac{1}{3}$
C$\frac{2}{5}$
D$\frac{2}{3}$
Show answer and explanation
Correct answer: D - $\frac{2}{3}$
The bag holds $6 + 5 + 4 = 15$ equally likely balls, and one ball cannot be both white and yellow, so the two probabilities add: $\frac{6}{15} + \frac{4}{15} = \frac{10}{15} = \frac{2}{3}$. The value $\frac{1}{3}$ is the chance of a black ball. The value $\frac{2}{5}$ counts the white balls alone, and $\frac{11}{15}$ adds the black balls in place of the yellow ones.
Q2
easy
Which of these numbers can never be the probability of an event?
A$0$
B$\frac{3}{7}$
C$-0.2$
D$1$
Show answer and explanation
Correct answer: C - $-0.2$
The probability of an event always lies from $0$ to $1$, both included, so a negative number such as $-0.2$ is impossible. The value $0$ is the probability of an impossible event and $1$ is the probability of a sure event, so both are allowed. The value $\frac{3}{7}$ lies between $0$ and $1$ and is an ordinary probability.
Q3
medium
The events $E_1, E_2, \ldots, E_n$ of a sample space $S$ are said to be exhaustive when which condition holds?
AEach $E_i$ contains exactly one sample point of $S$
B$P(E_1) = P(E_2) = \cdots = P(E_n)$
C$E_1 \cup E_2 \cup \cdots \cup E_n = S$
D$E_i \cap E_j = \phi$ for every pair with $i \neq j$
Exhaustive events together cover the whole sample space, so their union is $S$ and at least one of them is bound to occur. The condition on empty intersections defines mutually exclusive events, and events can be exhaustive while still overlapping. Events with one sample point each are elementary events, and equal probabilities describe equally likely events.
Q4
medium
Assertion (A): If $A$ and $B$ are mutually exclusive events with $P(A)>0$ and $P(B)>0$, then $P(A\mid B)=0$. Reason (R): Mutually exclusive events are always independent.
ABoth A and R are true, and R is the correct explanation of A
BBoth A and R are true, but R is not the correct explanation of A
CA is true, but R is false
DA is false, but R is true
Show answer and explanation
Correct answer: C - A is true, but R is false
(A) is true, because mutually exclusive events have $P(A\cap B)=0$ and so $P(A\mid B)=\frac{0}{P(B)}=0$. (R) is false, and (A) is the clearest evidence against it: independence would require $P(A\mid B)=P(A)$, which is positive here rather than $0$. Knowing that $B$ has occurred rules $A$ out completely, which is the opposite of leaving it unaffected.
Q5
easy
Following the axioms of probability, which statement holds for every event $E$ of a sample space $S$?
A$P(E) > 0$
B$0 \leq P(E) \leq 1$
C$P(E) \geq 1$
D$P(E)$ may be any real number
Show answer and explanation
Correct answer: B - $0 \leq P(E) \leq 1$
The axioms demand $P(E) \geq 0$ for every event together with $P(S) = 1$, and since $E$ is part of $S$ its probability cannot rise above 1, so every probability sits between 0 and 1. Strict positivity fails for the impossible event, whose probability is 0. Values below 0 or above 1 are forbidden outright, so the last two choices are wrong.
Q6
hard
For two events $A$ and $B$ it is known that $P(A)=0.3$ and $P(B)=0.5$. Which of the following values of $P(A\cap B)$ is impossible?
A$P(A\cap B)=0$
B$P(A\cap B)=0.4$
C$P(A\cap B)=0.3$
D$P(A\cap B)=0.15$
Show answer and explanation
Correct answer: B - $P(A\cap B)=0.4$
An intersection can never be more likely than either event that contains it, so $P(A\cap B)\le P(A)=0.3$ and the value $0.4$ cannot occur. A value of $0$ just means the events are mutually exclusive, which is allowed. The value $0.15$ is $P(A)\,P(B)$ and corresponds to independent events. The value $0.3$ means $A$ lies entirely inside $B$, which is possible because $P(A)\le P(B)$.
Practice the full Probability bank free
All 225 questions with timed practice, instant scoring, and explanations. Free forever.