Probability

A fair coin is flipped. What is the probability of getting heads?

• A0
• B0.25
• C0.5✓
• D1

Answer: A fair coin has two equally likely outcomes. P(heads) = 1/2 = 0.5.

A fair die is rolled. What is the probability of getting a 6?

• A1/5
• B1/6✓
• C1/4
• D1/3

Answer: A fair die has 6 equally likely outcomes. P(6) = 1/6.

A bag contains 3 red balls and 2 blue balls. If you pick one at random, what is the probability of picking red?

• A2/5
• B1/2
• C3/2
• D3/5✓

Answer: Total balls = 5. Red balls = 3. P(red) = 3/5.

A fair die is rolled. What is the probability of getting a number greater than 4?

• A2/6✓
• B3/6
• C1/6
• D4/6

Answer: Numbers greater than 4 are 5 and 6. P(>4) = 2/6 = 1/3.

A card is drawn from a standard deck (52 cards). What is the probability of drawing an ace?

• A12/52
• B4/52✓
• C1/52
• D13/52

Answer: There are 4 aces in a deck of 52 cards. P(ace) = 4/52 = 1/13.

A spinner has sections labelled 1, 2, 3, and 4. If you spin it twice, what is the probability of getting two 4s?

• A1/4
• B1/8
• C1/16✓
• D2/4

Answer: P(first 4) = 1/4. P(second 4) = 1/4. P(both 4) = 1/4 × 1/4 = 1/16.

A die is rolled 60 times. The number 3 appears 10 times. What is the relative frequency of rolling a 3?

• A1/10
• B1/60
• C10/60✓
• D3/10

Answer: Relative frequency = (number of times event occurs) ÷ (total number of trials) = 10/60 = 1/6.

If the probability of an event is 0.8, what is the probability it doesn't happen?

• A0.5
• B0.2✓
• C0.8
• D1.8

Answer: P(event doesn't happen) = 1 - P(event) = 1 - 0.8 = 0.2.

Two fair coins are flipped. What is the probability of getting at least one head?

• A3/4✓
• B1/2
• C1
• D1/4

Answer: Possible outcomes: HH, HT, TH, TT. 'At least one head' = HH, HT, TH = 3 outcomes. P = 3/4.

A bag contains 5 red, 3 blue, and 2 green balls. If you draw one at random, what is the probability of drawing green?

• A3/10
• B5/10
• C2/5
• D2/10✓

Answer: Total = 5 + 3 + 2 = 10. Green = 2. P(green) = 2/10 = 1/5.

A die is rolled. What is the probability of getting an even number?

• A1/3
• B2/3
• C1/2✓
• D1/6

Answer: Even numbers on a die: 2, 4, 6. That's 3 out of 6. P(even) = 3/6 = 1/2.

In a class of 30 students, 18 wear glasses. If you pick one student at random, what is the probability they wear glasses?

• A12/30
• B3/5✓
• C12/18
• D1/2

Answer: P(glasses) = 18/30 = 3/5.

A deck of cards is shuffled. What is the probability of drawing a red card?

• A1/2✓
• B1/52
• C1/26
• D13/52

Answer: Half the cards are red (hearts and diamonds). P(red) = 26/52 = 1/2.

A letter is chosen at random from the word 'PROBABILITY'. What is the probability it is the letter 'B'?

• A2/11✓
• B1/11
• C1/3
• D3/11

Answer: PROBABILITY has 11 letters. The letter B appears twice. P(B) = 2/11.

If P(A) = 0.6 and P(B) = 0.3, and events A and B are independent, what is P(A and B)?

• A0.18✓
• B0.3
• C0.6
• D0.9

Answer: For independent events, P(A and B) = P(A) × P(B) = 0.6 × 0.3 = 0.18.

A fair coin is flipped twice. What is the probability of getting two heads?

• A1/8
• B1/3
• C1/2
• D1/4✓

Answer: P(HH) = P(H) × P(H) = 1/2 × 1/2 = 1/4.

A fair die is rolled once. What is the probability of rolling a number greater than 4?

• A1/6
• B1/2
• C2/3
• D1/3✓

Answer: Numbers greater than 4 are 5 and 6. So P = 2/6 = 1/3.

A bag contains 3 red balls and 2 blue balls. If one ball is drawn, what is the probability it is red?

• A1/3
• B3/5✓
• C3/2
• D2/5

Answer: Total balls = 5. Red balls = 3. P(red) = 3/5.

Two fair dice are rolled. What is the probability that the sum is 7?

• A1/5
• B7/36
• C5/36
• D6/36✓

Answer: Pairs that sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes. Total outcomes = 36. P = 6/36 = 1/6.

A bag contains 4 red, 3 blue, and 2 green marbles. If two marbles are drawn without replacement, what is the probability both are red?

• A16/81
• B12/72✓
• C4/9
• D2/9

Answer: P(both red) = (4/9) × (3/8) = 12/72 = 1/6.

Events A and B are mutually exclusive. P(A) = 0.3 and P(B) = 0.2. What is P(A or B)?

• A0.06
• B0.3
• C0.5✓
• D0.6

Answer: For mutually exclusive events: P(A or B) = P(A) + P(B) = 0.3 + 0.2 = 0.5.

Events A and B are independent. P(A) = 0.4 and P(B) = 0.5. What is P(A and B)?

• A0.1
• B0.2✓
• C0.4
• D0.9

Answer: For independent events: P(A and B) = P(A) × P(B) = 0.4 × 0.5 = 0.2.

A Venn diagram shows two sets A and B. If P(A) = 0.6, P(B) = 0.5, and P(A and B) = 0.2, what is P(A or B)?

• A0.3
• B0.7
• C0.9✓
• D1.1

Answer: P(A or B) = P(A) + P(B) - P(A and B) = 0.6 + 0.5 - 0.2 = 0.9.

A frequency tree shows 100 people: 60 like tea, 40 like coffee. Of those who like tea, 35 also like coffee. How many like only tea?

• A15
• B25✓
• C45
• D55

Answer: Of the 60 who like tea, 35 also like coffee, so 60 - 35 = 25 like only tea.

A frequency tree shows 100 people: 60 like tea, 40 like coffee. Of those who like tea, 45 also like coffee. How many like only tea?

• A15✓
• B25
• C45
• D55

Answer: 60 like tea, 45 like both tea and coffee. Only tea = 60 - 45 = 15.

A spinner is spun 200 times and lands on red 50 times. What is the relative frequency of landing on red?

• A0.5
• B4
• C0.25✓
• D50

Answer: Relative frequency = number of times the outcome happened ÷ total number of trials = 50 ÷ 200 = 0.25. Dividing the other way round gives 4, and 50 is the raw count rather than a relative frequency.

A tree diagram shows first event with outcomes A (prob 0.3) and not A (prob 0.7). If A occurs, event B has prob 0.6. What is P(A and B)?

• A0.18✓
• B0.3
• C0.6
• D0.9

Answer: P(A and B) = P(A) × P(B|A) = 0.3 × 0.6 = 0.18.

A card is drawn from a standard deck (52 cards). What is the probability it is a heart?

• A1/13
• B1/26
• C1/2
• D1/4✓

Answer: There are 13 hearts in 52 cards. P = 13/52 = 1/4.

A spinner has sections numbered 1-6. It is spun twice. What is the probability of getting two 6s?

• A1/12
• B2/36
• C1/6
• D1/36✓

Answer: P(6 and 6) = (1/6) × (1/6) = 1/36.

A bag has 5 red and 3 blue balls. Two are drawn without replacement. What is the probability both are blue?

• A15/56
• B9/64
• C3/8
• D3/28✓

Answer: P(both blue) = (3/8) × (2/7) = 6/56 = 3/28.

Events A and B are such that P(A) = 0.6, P(B) = 0.4, and P(A or B) = 0.76. Are A and B independent?

• AYes, P(A and B) = 0.24✓
• BCannot determine
• CNo, P(A and B) = 0.18
• DNo, they are dependent

Answer: For independence, P(A and B) must equal P(A)×P(B) = 0.6 × 0.4 = 0.24. From P(A or B) = P(A) + P(B) - P(A and B): 0.76 = 0.6 + 0.4 - P(A and B), so P(A and B) = 0.24. Since 0.24 = P(A)×P(B), A and B are independent.

In a class of 30 pupils, 18 study French, 14 study German and 7 study both. How many study neither language?

• A2
• B7
• C5✓
• D11

Answer: On a Venn diagram the overlap holds 7, so French only holds 18 - 7 = 11 and German only holds 14 - 7 = 7. That accounts for 11 + 7 + 7 = 25 pupils, leaving 30 - 25 = 5 who study neither. Simply doing 30 - 18 - 14 counts the 7 twice.