A fair six-sided die is rolled once. Find the probability that the score is (a) even, (b) greater than 4, (c) a multiple of 3.
(a) \( \dfrac{3}{6} = \dfrac{1}{2} \). (b) \( \dfrac{2}{6} = \dfrac{1}{3} \). (c) \( \dfrac{2}{6} = \dfrac{1}{3} \).
Probability
10 exam-style questions on probability, tree diagrams, conditional probability and Venn diagrams. Tap Show answer for full working.
3 Easy4 Medium3 Hard
Two fair coins are tossed. Find the probability of (a) two heads, (b) at least one head, (c) exactly one head.
Sample space: HH, HT, TH, TT. (a) \( \dfrac{1}{4} \). (b) \( \dfrac{3}{4} \). (c) \( \dfrac{2}{4} = \dfrac{1}{2} \).
Events \( A \) and \( B \) are independent with \( P(A)=0.4 \) and \( P(B)=0.3 \). Find \( P(A \cap B) \) and \( P(A \cup B) \).
\( P(A \cap B)=0.4 \times 0.3=0.12 \). \( P(A \cup B)=0.4+0.3-0.12=0.58 \).
A bag contains 5 red and 3 blue discs. Two discs are drawn at random without replacement. Find the probability that (a) both are red, (b) they are different colours.
(a) \( \dfrac{5}{8} \times \dfrac{4}{7} = \dfrac{20}{56} = \dfrac{5}{14} \). (b) \( \dfrac{5}{8}\times\dfrac{3}{7} + \dfrac{3}{8}\times\dfrac{5}{7} = \dfrac{15}{56}+\dfrac{15}{56}=\dfrac{30}{56}=\dfrac{15}{28} \).
In a group of 40 students, 25 study Maths, 18 study Physics and 10 study both. A student is chosen at random. Find the probability that the student studies (a) Maths or Physics, (b) neither subject.
\( |M \cup P| = 25+18-10=33 \). (a) \( \dfrac{33}{40} \). (b) \( \dfrac{7}{40} \).
The probability that it rains on a given day is 0.3. If it rains, the probability that a bus is late is 0.4; if it does not rain, the probability the bus is late is 0.1. Find the probability that (a) the bus is late, (b) it rained given that the bus is late.
(a) \( P(L)=0.3\times0.4 + 0.7\times0.1 = 0.12+0.07=0.19 \). (b) \( P(R|L)=\dfrac{0.12}{0.19}=\dfrac{12}{19} \).
Events \( A \) and \( B \) satisfy \( P(A)=0.5 \), \( P(B)=0.6 \) and \( P(A \cup B)=0.8 \). Find \( P(A \cap B) \) and determine whether \( A \) and \( B \) are independent.
\( P(A \cap B)=0.5+0.6-0.8=0.3 \). \( P(A)P(B)=0.3 \), so yes, independent (since product equals intersection).
A factory has two machines. Machine A produces 60% of items and Machine B 40%. The probability an item from A is defective is 0.02; from B it is 0.05. An item is chosen at random and found to be defective. Find the probability it came from Machine B.
\( P(D)=0.6\times0.02 + 0.4\times0.05 = 0.012+0.02=0.032 \). \( P(B|D)=\dfrac{0.4\times0.05}{0.032}=\dfrac{0.02}{0.032}=\dfrac{5}{8} \).
Three cards are drawn at random without replacement from a standard pack of 52 cards. Find the probability that all three are hearts, and the probability that exactly two are hearts.
All hearts: \( \dfrac{13}{52}\times\dfrac{12}{51}\times\dfrac{11}{50}=\dfrac{1716}{132600}=\dfrac{11}{850} \). Exactly two: \( \dbinom{3}{2}\times\dfrac{13}{52}\times\dfrac{12}{51}\times\dfrac{39}{50}=\dfrac{3\times13\times12\times39}{52\times51\times50}=\dfrac{3\times13\times12\times39}{132600}=\dfrac{117}{850} \) (after simplifying).
The events \( A \) and \( B \) are such that \( P(A)=0.4 \), \( P(B|A)=0.5 \) and \( P(B|A')=0.3 \). Find \( P(B) \) and \( P(A|B) \).
\( P(B)=P(B|A)P(A)+P(B|A')P(A')=0.5\times0.4 + 0.3\times0.6=0.2+0.18=0.38 \). \( P(A|B)=\dfrac{P(B|A)P(A)}{P(B)}=\dfrac{0.2}{0.38}=\dfrac{10}{19} \).