Probability & distributions

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1. 1Probability rulesThe addition rule for 'or', mutually exclusive and independent events, several people each trying once ('only one', 'none', 'at least one'), and drawing with and without replacement.Done18 min
2. 2The binomial distributionRepeated independent trials with the same chance of success: P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ, choosing p and q correctly, and 'at least', 'at most' and 'more than' with the complement.Done18 min
3. 3The Poisson and normal distributionsThe Poisson distribution for rare events and as an approximation to the binomial, and the normal distribution with z-scores and tables.Done16 min

Past questions (75)

From 2008–2023: 69 WAEC, 6 NECO. Theory questions first, then objective, newest first.

Theory (65)

• NECO 2023 · Paper 2 · Q14A fair coin is tossed four times. Calculate the probability of obtaining (i) at least one tail; (ii) an equal number of heads and tails.
• NECO 2023 · Paper 2 · Q15The payoff matrix of players $A$ and $B$ in a game is $\begin{pmatrix} 3 & 4 \\ 5 & 1 \end{pmatrix}$. Find the (i) maximin value; (ii) minimax …
• WAEC 2023 · Paper 2 · Q12A farmer finds out that for every 100 oranges he harvests, 15 are bad. If he selects 20 oranges from his farm, what is the probability that three …
• WAEC 2023 · Paper 2 · Q13The mean of four consecutive odd numbers is 6. Find, correct to one decimal place, their variance.
• WAEC 2023 · Paper 2 · Q6In an examination, $80\%$ of the candidates passed. If 8 of the candidates are selected at random, calculate, correct to three decimal places, …
• WAEC 2022 · Paper 2 · Q12A basket contains 12 fruits: orange, apple and avocado pear, all of the same size. The numbers of oranges, apples and avocado pears form three …
• WAEC 2022 · Paper 2 · Q5The probability that Abiola will be late to office on a given day is $\frac25$. In a given working week of six days, find, correct to four significant …
• WAEC 2022 · Paper 2 · Q13A soldier fires at a target and the probability of hitting the target with any shot is $\frac25$. If he fires 6 shots, find, correct to three …
• WAEC 2022 · Paper 2 · Q6An objective test consists of 10 questions. A candidate is required to select the correct option from four options to every question. If the …
• WAEC 2020 · Paper 2 · Q12In a bottle manufacturing company, it was observed that $5\%$ of the bottles manufactured were defective. In a random sample of 150 bottles manufactured, …
• WAEC 2020 · Paper 2 · Q5Thirty percent ($30\%$) of the bulbs in a large box are defective. If $12$ bulbs are selected randomly from the box, calculate the probability …
• WAEC 2019 · Paper 2 · Q12In an examination, $60\%$ of the candidates passed. If 10 candidates are selected at random, find, correct to four decimal places, the probability …
• WAEC 2019 · Paper 2 · Q8Two fair dice are thrown together two times. Find the probability of obtaining a sum of seven in the first throw and a sum of four in the second …
• WAEC 2019 · Paper 2 · Q13In a bakery, $30\%$ of loaves of bread produced are of bad quality. If twelve loaves are selected at random from the bakery, calculate, correct …
• WAEC 2019 · Paper 2 · Q5Three soldiers, $X$, $Y$ and $Z$, have probabilities $\frac13$, $\frac15$ and $\frac14$ respectively of hitting a target. If each of them fires …
• WAEC 2019 · Paper 2 · Q13A fair coin is tossed 6 times. Calculate the probability of obtaining at most three heads.
• WAEC 2019 · Paper 2 · Q5A fair die with faces 1, 2, 3, 4, 5 and 6 is tossed twice. Calculate the probability that the sum of the numbers that show up is:
• WAEC 2018 · Paper 2 · Q12| Number of heads | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
• WAEC 2018 · Paper 2 · Q13The probabilities that Ali, Baba and Katty will gain admission to college are $\frac23$, $\frac34$ and $\frac45$ respectively. Find the probability …
• WAEC 2018 · Paper 2 · Q6The probability that Kunle solves a particular question is $\frac13$ while that of Tayo is $\frac15$. If both of them attempt the question, find …
• WAEC 2018 · Paper 2 · Q13At the end of full time, a football match between teams $A$ and $B$ was goalless. The winning team had to be decided through penalty kicks. Each …
• WAEC 2018 · Paper 2 · Q5The probability that a boy wins a race is 0.8. In four different races, what is the probability that he wins:
• WAEC 2018 · Paper 2 · Q13One out of every 3 bolts produced by a machine is defective. If 4 of the bolts produced by the machine are selected at random, find the probability …
• WAEC 2017 · Paper 2 · Q12In a school, the ratio of those who passed to those who failed a History test is $4 : 1$. If 7 students are selected at random from the school, …
• WAEC 2017 · Paper 2 · Q5Bottles of the same size produced in a factory are packed in boxes. Each box contains 10 bottles. If $8\%$ of the bottles produced are defective, …
• WAEC 2017 · Paper 2 · Q12| Marks | 0–2 | 3–5 | 6–8 | 9–11 | 12–14 | 15–17 |
• WAEC 2017 · Paper 2 · Q13A bag contains 10 identical balls of which 3 are black and the rest brown. Two balls are drawn at random one after the other from the bag without …
• WAEC 2017 · Paper 2 · Q5A committee of 5 members is to be formed from 6 men and 7 women. Calculate the probability that it consists of:
• WAEC 2016 · Paper 2 · Q13Two different Mathematics books, 5 different Physics books and 3 different Chemistry books are to be arranged on a shelf. How many arrangements …
• WAEC 2016 · Paper 2 · Q5The probabilities that Ago, Sulley and Musa will gain admission to a certain university are $\frac45$, $\frac34$ and $\frac23$ respectively. …
• WAEC 2016 · Paper 2 · Q12A survey indicated that $65\%$ of the families in a town own cars. Find, correct to three decimal places, the probability that among 7 families …
• WAEC 2016 · Paper 2 · Q13Using an assumed mean of 40, find the standard deviation of the following set of numbers: 28, 31, 32, 37, 40, 42, 45, 46, 48 and 50.
• WAEC 2016 · Paper 2 · Q5If ${}^nP_5 \div {}^nC_4 = 24$, find the value of $n$.
• WAEC 2014 · Paper 2 · Q13The probabilities that Kofi, Kwasi and Ama will pass a certain examination are $\frac{9}{10}$, $\frac45$ and $x$ respectively. If the probability …
• WAEC 2014 · Paper 2 · Q6A committee of 3 is formed from a panel of 5 men and 3 women. Find the:
• WAEC 2014 · Paper 2 · Q13A fair coin is tossed four times. Calculate the probability of obtaining: (i) at least one head; (ii) an equal number of heads and tails.
• WAEC 2014 · Paper 2 · Q5Events $A$ and $B$ are such that $P(A) = \frac13$ and $P(B) = \frac16$. Find $P(A \cup B)$ if events $A$ and $B$ are:
• WAEC 2013 · Paper 2 · Q14A bag contains 5 blue, 4 green and 3 yellow balls. All the balls are identical except for colour. Three balls are drawn at random without replacement. …
• WAEC 2013 · Paper 2 · Q15The probability that a man wins a race is 0.8. In four different races, what is the probability that he wins: (i) all races; (ii) no race; (iii) …
• WAEC 2013 · Paper 2 · Q5Three school prefects are to be chosen from four girls and five boys. What is the probability that:
• WAEC 2013 · Paper 2 · Q13A committee of 6 is to be selected at random from a group of 7 boys and 4 girls. (i) In how many ways can this be done if there are no restrictions? …
• WAEC 2012 · Paper 2 · Q13A hunter hits the target 3 times out of every five trials made. If 4 hunters aim at the target, calculate:
• WAEC 2012 · Paper 2 · Q14The marks scored by forty candidates in an examination are shown in the table.
• WAEC 2012 · Paper 2 · Q15A class consists of 6 girls and 10 boys. If a committee of 3 is chosen at random from the class, find the probability that:
• WAEC 2012 · Paper 2 · Q7| Number of heads | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
• WAEC 2011 · Paper 2 · Q13A survey conducted revealed that four out of every twenty taxi drivers do not have a valid driving licence. If 6 drivers are selected at …
• WAEC 2011 · Paper 2 · Q15A bag contains 4 red, 6 blue and 8 green identical marbles.
• WAEC 2011 · Paper 2 · Q13A committee of five is to be formed among 6 Ghanaians, 8 Nigerians and 5 Gambians. In how many ways can the committee be formed if: (i) there …
• WAEC 2011 · Paper 2 · Q15The probability that a patient recovers from a disease is 0.25. If 6 people are known to have contracted this disease, calculate the probability …
• WAEC 2011 · Paper 2 · Q5$X$ and $Y$ are two events such that $P(X \cup Y) = \frac{11}{15}$ and $P(X) = \frac13$. Find $P(Y)$ if the events are:
• WAEC 2011 · Paper 2 · Q6In a physics examination, the mean mark of the first twelve students in a class is 60, that of the next twenty students is 50 and that of the …
• WAEC 2010 · Paper 2 · Q14A box $P$ contains 3 white and 5 green identical balls. Another box $Q$ contains 6 white and 4 green identical balls. A ball is drawn at random …
• WAEC 2010 · Paper 2 · Q15A team of 3 students is to be selected from 5 boys and 4 girls to represent a school in a quiz. (i) If there is no restriction, in how many ways …
• WAEC 2010 · Paper 2 · Q5A committee of four persons is to be formed from 7 girls and 5 boys. Calculate, correct to two decimal places, the probability that the committee …
• WAEC 2010 · Paper 2 · Q6The table gives the distribution of heights, in metres, of 100 students of the same age group.
• WAEC 2009 · Paper 2 · Q13Simplify ${}^{n+1}C_3 - {}^{n-1}C_3$.
• WAEC 2009 · Paper 2 · Q5The probabilities of Rotey obtaining the highest mark in Mathematics, Physics and Biology tests are $0.9$, $0.75$ and $0.8$ respectively. Calculate …
• WAEC 2009 · Paper 2 · Q6A student representative council consists of 8 girls and 6 boys. If an editorial board of 5 persons is to be formed, what is the probability …
• WAEC 2009 · Paper 2 · Q13If ${}^nP_3 = 10\left({}^{n-2}P_2\right)$, find $n$.
• WAEC 2009 · Paper 2 · Q5The table shows the marks obtained by a group of students.
• WAEC 2009 · Paper 2 · Q6The chances of three hunters hitting a target are $\frac13$, $\frac23$ and $\frac15$ respectively. If they fire independently at the target, …
• WAEC 2008 · Paper 2 · Q14The following table shows the distribution of marks (%) obtained by some students in an examination.
• WAEC 2008 · Paper 2 · Q5In a hotel the breakfast menu is a choice between yam ($Y$) or plantain ($P$) or both. The Venn diagram shows the choices made by 25 guests of …
• WAEC 2008 · Paper 2 · Q14(i) In how many ways can eight boys be seated in a row? (ii) If two of the boys in (a)(i) cannot sit together, in how many ways can they be seated?
• WAEC 2008 · Paper 2 · Q15Simplify $^nC_r + {^nC_{r-1}}$.

Objective (10)

• NECO 2023 · Paper 1 · Q41The probabilities of three events $A$, $B$ and $C$ occurring are $\frac13$, $\frac12$ and $\frac23$ respectively. Find the probability that only …
• NECO 2023 · Paper 1 · Q42One out of a thousand bulbs manufactured by a factory is found to be defective. If 2,000 bulbs were produced, find the probability that at least …
• NECO 2023 · Paper 1 · Q44A fair die is rolled once. What is the probability of obtaining a multiple of 2 or 3?
• NECO 2023 · Paper 1 · Q47In an examination, $75\%$ of the candidates passed. Use the binomial distribution to calculate the probability that a random sample of 8 candidates …
• WAEC 2023 · Paper 1 · Q15The probabilities that Atta and Tunde will hit a target in a shooting contest are $\frac{1}{6}$ and $\frac{1}{9}$ respectively. Find the probability …
• WAEC 2023 · Paper 1 · Q31If $X$ and $Y$ are two independent events such that $P(X) = \frac{1}{8}$ and $P(X \cup Y) = \frac{5}{8}$, find $P(Y)$.
• WAEC 2022 · Paper 1 · Q22The table shows the distribution of the distance (in km) covered by 40 hunters while hunting.
• WAEC 2022 · Paper 1 · Q32The probability that a student will graduate from a college is 0.4. If 3 students are selected from the college, what is the probability …
• WAEC 2020 · Paper 1 · Q17Calculate the probability that the product of two numbers selected at random with replacement from the set $\{-5, -2, 4, 8\}$ is positive.
• WAEC 2020 · Paper 1 · Q34A bag contains 5 red and 5 blue identical balls. Three balls are selected at random without replacement. Determine the probability of selecting …