Secondary school › Counting and probability

Probability

Sample space and events, classical and geometric probability, the addition rule and the complement. 9 problems with step-by-step solutions.

Solve each problem in your notebook first, then open the solution.

Problem 1 Sample space and events

a) Write the sample space for tossing a coin twice, and for a family with 3 children (B = boy, G = girl, oldest first).

b) The sample space is \(\{1, 2, \dots, 12\}\). Write the event "the number is prime" and the event "the number is divisible by 3".

c) A die is rolled, \(A = \{2, 3, 4\}\), \(B = \{3, 4, 5, 6\}\). Find \(A\cup B\), \(A\cap B\) and \(A'\).

Show solution

a) \(\{HH, HT, TH, TT\}\). \(\{BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG\}\).

b) \(\{2, 3, 5, 7, 11\}\); \(\{3, 6, 9, 12\}\).

c) \(\{2, 3, 4, 5, 6\}\); \(\{3, 4\}\); \(\{1, 5, 6\}\).

Problem 2 Classical probability

\(P(E) = \dfrac{n(E)}{n(S)}\).

a) A die is rolled. Find the probability of a number greater than 4.

b) The numbers 1 to 20 are on cards. Find the probability of a prime number.

c) A coin is tossed 3 times. Find the probability of exactly two heads.

d) A card is drawn from 52. Find the probability of a king, and of a red card.

Show solution

a) \(\frac26 = \frac13\).

b) 8 primes (2, 3, 5, 7, 11, 13, 17, 19): \(\frac{8}{20} = \frac25\).

c) \(\{HHT, HTH, THH\}\): \(\frac38\).

d) \(\frac{4}{52} = \frac{1}{13}\); \(\frac{26}{52} = \frac12\).

Problem 3 Two and three dice

a) Two dice are rolled. Find the probability that the sum is 8, and that the sum is less than 5.

b) Which sum of two dice is the most likely? Explain.

c) Three dice are rolled. Find the probability that the sum is 16.

Show solution

a) Sum 8: (2,6), (3,5), (4,4), (5,3), (6,2): \(\frac{5}{36}\). Sum less than 5: sums 2, 3, 4 give \(1 + 2 + 3 = 6\) outcomes: \(\frac{6}{36} = \frac16\).

b) Sum 7: it has 6 outcomes out of 36, more than any other sum.

c) Sum 16: (6,6,4) in 3 orders and (6,5,5) in 3 orders: \(\frac{6}{216} = \frac{1}{36}\).

Problem 4 Certain and impossible events

a) A die is rolled. Find the probability that the number is less than 7, and that it is 8.

b) A die has 2 white, 1 red and 3 green faces. Find the probability that the top face is green.

c) A box has 20 bulbs and 3 of them are broken. One bulb is chosen. Find the probability that it works.

Show solution

a) Certain event: \(1\). Impossible event: \(0\).

b) \(\frac36 = \frac12\).

c) \(\frac{17}{20}\).

Problem 5 Geometric probability

A dart board has circles with radii 2 cm, 4 cm and 6 cm with a common centre. A dart hits the board. Find the probability that it lands in the smallest circle, and in the outer ring.

Show solution

Probability = ratio of areas. Board area \(36\pi\).

Smallest circle: \(\frac{4\pi}{36\pi} = \frac19\). Outer ring: \(\frac{36\pi - 16\pi}{36\pi} = \frac59\).

Problem 6 Addition rule

\(P(A\cup B) = P(A) + P(B) - P(A\cap B)\).

a) A die is rolled. Find the probability of an odd number or a number greater than 4.

b) A box has 4 blue, 6 red and 5 yellow marbles. Find the probability of a red or yellow marble.

c) Cards 1–30. Find the probability of a number divisible by 3 or by 5.

d) A number is chosen from 1–100. Find the probability that it is divisible by both 3 and 4, and by 3 or 4.

Show solution

a) Odd \(\{1, 3, 5\}\), greater than 4 \(\{5, 6\}\), both \(\{5\}\): \(\frac{3 + 2 - 1}{6} = \frac23\).

b) \(\frac{6 + 5}{15} = \frac{11}{15}\).

c) \(\frac{10 + 6 - 2}{30} = \frac{14}{30} = \frac{7}{15}\).

d) Both (divisible by 12): \(\frac{8}{100} = \frac{2}{25}\). Either: \(\frac{33 + 25 - 8}{100} = \frac12\).

Problem 7 Complement

\(P(A') = 1 - P(A)\).

a) A coin is tossed 5 times. Find the probability of at least one tail.

b) 5 people are chosen at random. Find the probability that at least two of them have the birthday in the same month (12 equally likely months).

c) A school has 400 pupils: 120 play football, 80 basketball and 40 both. A pupil is chosen at random. Find the probability that he plays neither.

Show solution

a) \(1 - \frac{1}{32} = \frac{31}{32}\).

b) All different months: \(\frac{12\cdot 11\cdot 10\cdot 9\cdot 8}{12^5} = \frac{95040}{248832} \approx 0.382\). Answer \(\approx 0.618\).

c) Football or basketball: \(120 + 80 - 40 = 160\). Neither: \(\frac{240}{400} = \frac35\).

Problem 8 Counting in probability

a) A three-digit number with different digits is formed from \(\{1, 2, 3, 4, 5, 6\}\). Find the probability that it is odd.

b) Six people, including Ana and Bojan, sit at random around a round table. Find the probability that Ana and Bojan sit together.

c) A box has 5 yellow and 7 red marbles. Two are drawn. Find the probability that both are red.

d) A box has 15 bulbs, 4 broken. Three are chosen. Find the probability that exactly one is broken, and that at least one is broken.

Show solution

a) The last digit is 1, 3 or 5: \(\frac36 = \frac12\).

b) Ana sits anywhere; Bojan has 5 seats, 2 next to Ana: \(\frac25\).

c) \(\frac{C(7, 2)}{C(12, 2)} = \frac{21}{66} = \frac{7}{22}\).

d) Exactly one: \(\frac{C(4, 1)C(11, 2)}{C(15, 3)} = \frac{220}{455} = \frac{44}{91}\). At least one: \(1 - \frac{C(11, 3)}{455} = 1 - \frac{165}{455} = \frac{58}{91}\).

Problem 9 Counting in probability

a) A coin is tossed 10 times. Find the probability of exactly 6 heads.

b) A committee of 4 is chosen from 5 boys and 6 girls. Find the probability of 2 boys and 2 girls.

c) A pupil guesses all 10 answers of a true/false test. Find the probability of exactly 8 correct.

d) 8 people sit in a row at random. Find the probability that two given people sit next to each other.

Show solution

a) \(\frac{C(10, 6)}{2^{10}} = \frac{210}{1024} = \frac{105}{512}\).

b) \(\frac{C(5, 2)C(6, 2)}{C(11, 4)} = \frac{150}{330} = \frac{5}{11}\).

c) \(\frac{C(10, 8)}{2^{10}} = \frac{45}{1024}\).

d) \(\frac{2\cdot 7!}{8!} = \frac14\).