Write the sample space and calculate the probability based on the given information.
- (i) Two coins are tossed at the same time. What is the probability of getting at least one head?
- (ii) Ten identical cards numbered 1 to 10 are placed in a box. One card is drawn at random. What is the probability of drawing a card with an even number?
- (iii) A die is rolled once. What is the probability of getting a number greater than 4?
- (iv) A bag contains 3 red balls, 2 blue balls, and 1 green ball. One ball is picked at random. What is the probability that it is not red?
- (v) Three coins are tossed simultaneously. What is the probability of getting exactly two heads?
Step-by-step solution
Idea: For each part: write S (every outcome once), pick out the outcomes that fit the event, and use P = favourable outcomespossible outcomes.
(i) Two coins are tossed at the same time. What is the probability of getting at least one head?
- S = {HH, HT, TH, TT}, n(S) = 4.½ mark
- “At least one head” means one or two heads: {HH, HT, TH}, 3 outcomes. P = 34.½ mark
(ii) Ten identical cards numbered 1 to 10 are placed in a box. One card is drawn at random. What is the probability of drawing a card with an even number?
- S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, n(S) = 10.½ mark
- Even numbers: {2, 4, 6, 8, 10}, 5 outcomes. P = 510 = 12.½ mark
(iii) A die is rolled once. What is the probability of getting a number greater than 4?
- S = {1, 2, 3, 4, 5, 6}, n(S) = 6.½ mark
- Greater than 4: {5, 6} (4 itself is not greater than 4). P = 26 = 13.½ mark
(iv) A bag contains 3 red balls, 2 blue balls, and 1 green ball. One ball is picked at random. What is the probability that it is not red?
- Label the balls: S = {R1, R2, R3, B1, B2, G}, n(S) = 6.½ mark
- Not red: {B1, B2, G}, 3 outcomes. P = 36 = 12.½ mark
(v) Three coins are tossed simultaneously. What is the probability of getting exactly two heads?
- S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}, n(S) = 2 × 2 × 2 = 8.½ mark
- Exactly two heads: {HHT, HTH, THH}, 3 outcomes (HHH has three heads, so it is not included). P = 38.½ mark
Check: (i) P(no head) = P(TT) = 14 and 14 + 34 = 1 ✓. (iv) P(red) = 36, and 36 + 36 = 1 ✓. (v) Counts of heads 0, 1, 2, 3 occur 1, 3, 3, 1 times: total 8 ✓.
Answer to write in the exam
(i)
S = {HH, HT, TH, TT}; n(S) = 4
E = {HH, HT, TH}; n(E) = 3
∴ P(at least one head) = 34
(ii)
S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}; n(S) = 10
E = {2, 4, 6, 8, 10}; n(E) = 5
∴ P(even number) = 510 = 12
(iii)
S = {1, 2, 3, 4, 5, 6}; n(S) = 6
E = {5, 6}; n(E) = 2
∴ P(number greater than 4) = 26 = 13
(iv)
S = {R1, R2, R3, B1, B2, G}; n(S) = 6
E (not red) = {B1, B2, G}; n(E) = 3
∴ P(not red) = 36 = 12
(v)
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}; n(S) = 8
E = {HHT, HTH, THH}; n(E) = 3
∴ P(exactly two heads) = 38
Common mistakes that cost marks
- In (i), counting only HT and TH (exactly one head). “At least one” includes HH.
- In (iii), including 4 in “greater than 4” and getting 36.
- In (v), writing the sample space with only 4 outcomes, or counting HHH as “two heads”. Three coins give 8 outcomes, and exactly two means not three.
How this can come in the exam
Three coins are tossed together. The probability of getting no head is
- 0
- 18
- 13
- 38
Show answer
(B) 18
Only TTT out of 8 outcomes: 18.
Cards numbered 1 to 15 are put in a box and one is drawn at random. Write the sample space and find the probability that the number is a multiple of 4.
Show answer
S = {1, 2, …, 15}, n(S) = 15. Multiples of 4: {4, 8, 12}, 3 outcomes. P = 315 = 15.Try one yourself
A bag has 4 yellow, 1 white and 3 black balls. One ball is picked at random. Find the probability that it is not black.
Show answer
n(S) = 8; not black = 4 + 1 = 5 balls. P = 58.
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