A fair coin is tossed once. Write down the probability that it lands heads.
(1)
2
A six-sided die is rolled once. State the probability of rolling a 4.
(1)
3
A bag contains 5 red counters and 3 blue counters. A counter is chosen at random. Give the probability the counter is blue.
(1)
4
A student tosses a coin 50 times and gets 28 heads. Calculate the relative frequency of heads from this experiment.
(2)
5
A spinner with 8 equal sections is spun 40 times. The number of times it landed on section A was 6. Find the expected frequency for section A and compare to the experimental frequency. State which is larger.
(2)
6
A fair coin is tossed 100 times. What is the expected number of tails?
(2)
7
A die is rolled 120 times. The face 6 appears 18 times. Calculate the relative frequency of rolling a 6 and compare it to the theoretical probability. Which is higher?
(2)
8
A class carries out an experiment rolling a spinner with 5 equal sectors. After 75 spins, sector B came up 20 times. Calculate the relative frequency of B and give it as a fraction in simplest form.
(2)
9
A fair six-sided die is rolled 60 times. Using expected frequency, how many times would you expect to roll a number less than 4 (that is 1, 2 or 3)?
(2)
10
A coin is biased so that heads has probability 0.6 and tails 0.4. In 50 tosses, find the expected number of heads.
(2)
11
A spinner has 3 equal colours: red, green and yellow. A pupil spins it 90 times and records red 28 times. Calculate the relative frequency of red and find the difference between experimental and expected frequencies (experimental minus expected).
(2)
12
You flip a two-sided coin 20 times and get 12 heads. Another student flips a different coin 20 times and gets 8 heads. Which experimental result is closer to the expected frequency and by how many flips? Show working.
(2)
13
A die is rolled 30 times. The recorded counts for faces 1 to 6 are: 6, 4, 9, 3, 5, 3. Which face deviates most from the expected frequency and by how many rolls?
(3)
14
A spinner with 4 equal sections is spun 48 times and lands on sector C 18 times. Find the expected frequency for sector C and state how many more times it appeared than expected.
(2)
15
A biased coin has probability of heads 0.7. It is tossed 10 times and 9 heads are observed. Calculate the relative frequency of heads, the expected number of heads, and the difference experimental minus expected.
(3)
16
A class of 24 students each flips a fair coin twice, giving 48 tosses in total. They observe 30 heads. Find the relative frequency of heads, the expected number of heads, and say how many more heads were observed than expected.
(3)
17
Design a simple experiment to test whether a coin is fair. Give the main steps you would take and explain briefly what result would suggest the coin is fair.
(4)
18
A die is rolled 600 times and the face 6 appears 110 times. Calculate the experimental probability of rolling a 6 as a fraction and as a decimal. Give the theoretical probability and find the difference (experimental minus theoretical).
(4)
Mark scheme · KS3.M-P4 Experimental Probability and Relative Frequency
Question 1
B1 1/2 or 0.5 or 50% cao
Answer: 1/2
Question 2
B1 1/6 or about 0.166... or 16.7% cao
Answer: 1/6
Question 3
B1 3/8 cao
Answer: 3/8
Question 4
M1 correct method: heads divided by total 28/50 or equivalent
A1 0.56 or 56% cao
Answer: 0.56
Question 5
M1 expected frequency calculated as 1/8 of 40 = 40/8 or 5
A1 expected = 5, experimental = 6 so experimental is larger cao
Answer: 5; experimental 6, experimental is larger
Question 6
M1 method: 1/2 of 100 or 100 x 0.5
A1 50 cao
Answer: 50
Question 7
M1 relative frequency shown as 18/120 = 0.15 or equivalent
A1 theoretical is 1/6 = 0.166... so theoretical is larger cao
Answer: 0.15; theoretical 1/6 (about 0.166...), theoretical is larger
Question 8
M1 method: 20/75 or equivalent shown
A1 4/15 cao
Answer: 4/15
Question 9
M1 method: probability 3/6 = 1/2 then expected frequency 1/2 of 60
A1 30 cao
Answer: 30
Question 10
M1 method: 50 x 0.6
A1 30 cao
Answer: 30
Question 11
M1 relative frequency 28/90 = 14/45 or 0.311... shown
A1 expected 90/3 = 30 so difference = 28 - 30 = -2 cao
Answer: 14/45; difference -2
Question 12
M1 method: expected for 20 flips is 10 heads and differences calculated |12-10| and |8-10|
Answer: 0.625; expected 24; experimental higher by 6
Question 17
M1 suggest carrying out a large number of tosses, e.g. at least 50 or 100
M1 state recording outcomes and calculating relative frequency of heads
M1 state comparing relative frequency to theoretical probability 0.5
A1 conclude that a result close to 0.5 after many trials suggests the coin is fair cao
Answer: Do at least 50 to 100 tosses, record and calculate relative frequency of heads, compare to 0.5; a result close to 0.5 suggests the coin is fair
Question 18
M1 experimental probability as fraction 110/600 simplified to 11/60
M1 convert to decimal 11/60 = 0.183333... shown
M1 theoretical probability 1/6 = 0.166666... stated
A1 difference 11/60 - 1/6 = 1/60 (about 0.016666...) cao