Probability scale and relative frequency Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Probability scale and relative frequency questions. See exactly how to solve problems on theoretical probability, equally likely outcomes, simplifying fractions, fair spinner.

theoretical probabilityequally likely outcomessimplifying fractionsfair spinnerP(not A) = 1 - P(A)probabilities sum to 1
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
A bag contains 33 red counters, 55 blue counters and 22 green counters. Amina takes one counter at random. Work out the probability that the counter is red. Give your answer as a fraction in its simplest form.

Worked solution

  1. Count how many counters there are altogether

    3+5+2=103 + 5 + 2 = 10

    There are 33 red, 55 blue and 22 green counters, so there are 1010 counters in the bag. Taking one at random makes all 1010 counters equally likely, and that is what lets us use theoretical probability.

  2. Write the theoretical probability of red

    P(red)=310P(\text{red}) = \frac{3}{10}

    Theoretical probability is the number of favourable outcomes divided by the total number of equally likely outcomes, so P(red)=310P(\text{red}) = \frac{3}{10}.

  3. State the answer

    310\frac{3}{10}

    The probability is 310\frac{3}{10}, written as a fraction in its simplest form.

Answer
310\frac{3}{10}
Question 2
1 markeasy
A bag contains 44 red counters, 66 blue counters and 22 green counters. Ben takes one counter at random. Work out the probability that the counter is blue. Give your answer as a fraction in its simplest form.

Worked solution

  1. Count how many counters there are altogether

    4+6+2=124 + 6 + 2 = 12

    There are 44 red, 66 blue and 22 green counters, so there are 1212 counters in the bag. Taking one at random makes all 1212 counters equally likely, and that is what lets us use theoretical probability.

  2. Write the theoretical probability of blue

    P(blue)=612=12P(\text{blue}) = \frac{6}{12} = \frac{1}{2}

    Theoretical probability is the number of favourable outcomes divided by the total number of equally likely outcomes, so P(blue)=12P(\text{blue}) = \frac{1}{2}.

  3. State the answer

    12\frac{1}{2}

    The probability is 12\frac{1}{2}, written as a fraction in its simplest form.

Answer
12\frac{1}{2}
Question 3
2 markseasy
A bag contains 55 red counters, 33 blue counters and 44 green counters. Chloe takes one counter at random. Work out the probability that the counter is green. Give your answer as a fraction in its simplest form.

Worked solution

  1. Count how many counters there are altogether

    5+3+4=125 + 3 + 4 = 12

    There are 55 red, 33 blue and 44 green counters, so there are 1212 counters in the bag. Taking one at random makes all 1212 counters equally likely, and that is what lets us use theoretical probability.

  2. Write the theoretical probability of green

    P(green)=412=13P(\text{green}) = \frac{4}{12} = \frac{1}{3}

    Theoretical probability is the number of favourable outcomes divided by the total number of equally likely outcomes, so P(green)=13P(\text{green}) = \frac{1}{3}.

  3. State the answer

    13\frac{1}{3}

    The probability is 13\frac{1}{3}, written as a fraction in its simplest form.

Answer
13\frac{1}{3}
Question 4
1 markeasy
A fair spinner has 88 equal sections. 33 of the sections are red. The spinner is spun once. Work out the probability that it lands on red. Give your answer as a fraction in its simplest form.

Worked solution

  1. Count the equally likely outcomes

    sections=8\text{sections} = 8

    The spinner has 88 equal sections, so all 88 outcomes are equally likely. Equal sections are what makes theoretical probability usable here.

  2. Write the theoretical probability and simplify it

    P(red)=38P(\text{red}) = \frac{3}{8}

    Theoretical probability is favourable outcomes divided by total outcomes: 38\frac{3}{8}, which is 38\frac{3}{8} in its simplest form.

  3. State the answer

    38\frac{3}{8}

    The probability that the spinner lands on red is 38\frac{3}{8}.

Answer
38\frac{3}{8}
Question 5
2 markseasy
A fair spinner has 1010 equal sections. 44 of the sections are blue. The spinner is spun once. Work out the probability that it lands on blue. Give your answer as a fraction in its simplest form.

Worked solution

  1. Count the equally likely outcomes

    sections=10\text{sections} = 10

    The spinner has 1010 equal sections, so all 1010 outcomes are equally likely. Equal sections are what makes theoretical probability usable here.

  2. Write the theoretical probability and simplify it

    P(blue)=410=25P(\text{blue}) = \frac{4}{10} = \frac{2}{5}

    Theoretical probability is favourable outcomes divided by total outcomes: 410\frac{4}{10}, which is 25\frac{2}{5} in its simplest form.

  3. State the answer

    25\frac{2}{5}

    The probability that the spinner lands on blue is 25\frac{2}{5}.

Answer
25\frac{2}{5}

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