Conditional probability Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Conditional probability questions. See exactly how to solve problems on conditional probability, two-way tables, restricting the sample space, probability from a two-way table.

conditional probabilitytwo-way tablesrestricting the sample spaceprobability from a two-way tablereading frequenciessimplifying fractions
GCSE Higher70 questionsStep-by-step solutions
Question 1
2 markseasy
2424 students took part in a survey about how they travel to school. The students are either girls or boys. Some of the students walk to school and the rest do not walk to school. 77 of the students are girls who walk to school. 55 of the students are girls who do not walk to school. 44 of the students are boys who walk to school. 88 of the students are boys who do not walk to school. One of the 2424 students is chosen at random. Given that this student is one of the girls, work out the probability that this student is also one of the students who walk to school. Give your answer as a fraction in its simplest form.

Worked solution

  1. Restrict the universe to the group you are choosing from

    n(girls)=7+5=12n(\text{girls}) = 7 + 5 = 12

    You are told the student is one of the girls. Every other student in the survey is now irrelevant: the choice is made from the 1212 girls only. So 1212 is the denominator — NOT 2424.

  2. Count how many of that group are in the event, then write the fraction

    P=712=712P = \frac{7}{12} = \frac{7}{12}

    Of those 1212 girls, 77 walk to school — they are the 77 girls who walk to school. So the probability is 712=712\frac{7}{12} = \frac{7}{12}.

  3. State the answer

    712\frac{7}{12}

    So given that the student is one of the girls, the probability that the student is also one of the students who walk to school is 712\frac{7}{12}.

Answer
712\frac{7}{12}
Question 2
2 markseasy
2222 pupils took part in a survey about a spelling test. The pupils are either boys or girls. Some of the pupils passed the test and the rest did not pass the test. 66 of the pupils are boys who passed the test. 44 of the pupils are boys who did not pass the test. 99 of the pupils are girls who passed the test. 33 of the pupils are girls who did not pass the test. One of the pupils is chosen at random from those who passed the test. Work out the probability that this pupil is one of the girls. Give your answer as a fraction in its simplest form.

Worked solution

  1. Restrict the universe to the group you are choosing from

    n(pupils who passed the test)=6+9=15n(\text{pupils who passed the test}) = 6 + 9 = 15

    You are told the pupil is one of the pupils who passed the test. Every other pupil in the survey is now irrelevant: the choice is made from the 1515 pupils who passed the test only. So 1515 is the denominator — NOT 2222.

  2. Count how many of that group are in the event, then write the fraction

    P=915=35P = \frac{9}{15} = \frac{3}{5}

    Of those 1515 pupils who passed the test, 99 are girls — they are the 99 girls who passed the test. So the probability is 915=35\frac{9}{15} = \frac{3}{5}.

  3. State the answer

    35\frac{3}{5}

    So given that the pupil is one of the pupils who passed the test, the probability that the pupil is also one of the girls is 35\frac{3}{5}.

Answer
35\frac{3}{5}
Question 3
2 markseasy
3030 students took part in a survey about wearing glasses. The students are either girls or boys. Some of the students wear glasses and the rest do not wear glasses. 55 of the students are girls who wear glasses. 1010 of the students are girls who do not wear glasses. 88 of the students are boys who wear glasses. 77 of the students are boys who do not wear glasses. One of the 3030 students is chosen at random. Given that this student is one of the boys, work out the probability that this student is also one of the students who wear glasses. Give your answer as a fraction in its simplest form.

Worked solution

  1. Restrict the universe to the group you are choosing from

    n(boys)=8+7=15n(\text{boys}) = 8 + 7 = 15

    You are told the student is one of the boys. Every other student in the survey is now irrelevant: the choice is made from the 1515 boys only. So 1515 is the denominator — NOT 3030.

  2. Count how many of that group are in the event, then write the fraction

    P=815=815P = \frac{8}{15} = \frac{8}{15}

    Of those 1515 boys, 88 wear glasses — they are the 88 boys who wear glasses. So the probability is 815=815\frac{8}{15} = \frac{8}{15}.

  3. State the answer

    815\frac{8}{15}

    So given that the student is one of the boys, the probability that the student is also one of the students who wear glasses is 815\frac{8}{15}.

Answer
815\frac{8}{15}
Question 4
2 markseasy
3030 members took part in a survey at a sports club. The members are either men or women. Some of the members go swimming and the rest do not go swimming. 99 of the members are men who go swimming. 66 of the members are men who do not go swimming. 44 of the members are women who go swimming. 1111 of the members are women who do not go swimming. One of the members is chosen at random from those who do not go swimming. Work out the probability that this member is one of the men. Give your answer as a fraction in its simplest form.

Worked solution

  1. Restrict the universe to the group you are choosing from

    n(members who do not go swimming)=6+11=17n(\text{members who do not go swimming}) = 6 + 11 = 17

    You are told the member is one of the members who do not go swimming. Every other member in the survey is now irrelevant: the choice is made from the 1717 members who do not go swimming only. So 1717 is the denominator — NOT 3030.

  2. Count how many of that group are in the event, then write the fraction

    P=617=617P = \frac{6}{17} = \frac{6}{17}

    Of those 1717 members who do not go swimming, 66 are men — they are the 66 men who do not go swimming. So the probability is 617=617\frac{6}{17} = \frac{6}{17}.

  3. State the answer

    617\frac{6}{17}

    So given that the member is one of the members who do not go swimming, the probability that the member is also one of the men is 617\frac{6}{17}.

Answer
617\frac{6}{17}
Question 5
2 markseasy
3434 people took part in a survey about owning a pet. The people are either adults or children. Some of the people own a pet and the rest do not own a pet. 1212 of the people are adults who own a pet. 88 of the people are adults who do not own a pet. 55 of the people are children who own a pet. 99 of the people are children who do not own a pet. One of the 3434 people is chosen at random. Given that this person is one of the adults, work out the probability that this person is also one of the people who own a pet. Give your answer as a fraction in its simplest form.

Worked solution

  1. Restrict the universe to the group you are choosing from

    n(adults)=12+8=20n(\text{adults}) = 12 + 8 = 20

    You are told the person is one of the adults. Every other person in the survey is now irrelevant: the choice is made from the 2020 adults only. So 2020 is the denominator — NOT 3434.

  2. Count how many of that group are in the event, then write the fraction

    P=1220=35P = \frac{12}{20} = \frac{3}{5}

    Of those 2020 adults, 1212 own a pet — they are the 1212 adults who own a pet. So the probability is 1220=35\frac{12}{20} = \frac{3}{5}.

  3. State the answer

    35\frac{3}{5}

    So given that the person is one of the adults, the probability that the person is also one of the people who own a pet is 35\frac{3}{5}.

Answer
35\frac{3}{5}

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