Mathematics — CBSE Class 10 board question

Sample Space: When three coins are tossed, total outcomes = 8
{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

(a) Exactly two tails:
Favourable outcomes: {HTT, THT, TTH} → 3 outcomes

$$P(\text{exactly two tails}) = \frac{3}{8}$$

(b) At least one head:
Favourable outcomes: all except {TTT} → 7 outcomes

$$P(\text{at least one head}) = \frac{7}{8}$$

(c) At most two heads:
Favourable outcomes: all except {HHH} → 7 outcomes

$$P(\text{at most two heads}) = \frac{7}{8}$$

Source: Chapter 14, Section 14.1

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• Always list the sample space first — it shows the examiner you've correctly identified all 8 outcomes.
• "Exactly two tails" means precisely 2 tails, not more.
• "At least one head" means 1 or more heads; easiest via complement: 1 − P(no head) = 1 − 1/8 = 7/8.
• "At most two heads" means 0, 1, or 2 heads; complement is {HHH} (three heads), so 1 − 1/8 = 7/8.
• Each part carries 1 mark: one correct fraction with minimal working is sufficient.