Three fair coins are tossed simultaneously. If $X$ denotes the number of heads obtained, write the probability distribution of $X$.

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Accepted Answer

**Step 1 — Sample space.** Tossing three coins gives $2^3 = 8$ equally likely outcomes:
$$\{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$$

**Step 2 — Count heads in each outcome and group by value of $X$.**
- $X=0$ (no heads): $TTT$ → 1 outcome
- $X=1$ (one head): $HTT, THT, TTH$ → 3 outcomes
- $X=2$ (two heads): $HHT, HTH, THH$ → 3 outcomes
- $X=3$ (three heads): $HHH$ → 1 outcome

**Step 3 — Convert counts to probabilities** (each outcome has probability $	frac18$):
$$P(X=0)=	frac18,\quad P(X=1)=	frac38,\quad P(X=2)=	frac38,\quad P(X=3)=	frac18$$

**Dual-solve check (independent method).** Since each coin toss is an independent Bernoulli trial with $p=q=	frac12$, $X$ also follows a Binomial distribution $B\!\left(3, 	frac12ight)$, so
$$P(X=r) = \binom{3}{r}\left(	frac12ight)^r\left(	frac12ight)^{3-r} = \binom{3}{r}	frac18$$
giving $\binom30	frac18=	frac18$, $\binom31	frac18=	frac38$, $\binom32	frac18=	frac38$, $\binom33	frac18=	frac18$ — identical to Step 3, confirming the distribution.

**Step 4 — Validity check.** $	frac18+	frac38+	frac38+	frac18 = 1$ and every probability is non-negative, so this is a valid probability distribution.

> [!ANSWER]
> | $X$ | 0 | 1 | 2 | 3 |
> |---|---|---|---|---|
> | $P(X)$ | $	frac18$ | $	frac38$ | $	frac38$ | $	frac18$ |

Q.Three fair coins are tossed simultaneously. If $X$ denotes the number of heads obtained, write the probability distribution of $X$ .

Concept understanding — Random Variable and Probability Mass Function

Q.Three fair coins are tossed simultaneously. If XXX denotes the number of heads obtained, write the probability distribution of XXX.

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Q.Three fair coins are tossed simultaneously. If $X$ denotes the number of heads obtained, write the probability distribution of $X$ .

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