A baby cries if and only if she is hungry or tired, or both.

Let
𝐻
=
the baby is hungry
,
𝑇
=
the baby is tired
;
𝐶
=
the baby is crying

Assume:
𝑃
(
𝐻
)
=
ℎ
,
𝑃
(
𝑇
)
=
𝑡
,
0

Question

A baby cries if and only if she is hungry or tired, or both.

Let 
𝐻
=
the baby is hungry
, 
𝑇
=
the baby is tired
; 
𝐶
=
the baby is crying

Assume: 
𝑃
(
𝐻
)
=
ℎ
,
𝑃
(
𝑇
)
=
𝑡
,
0
<
ℎ
,
𝑡
<
1

and assume that (H) and (T) are independent.

Answer the following question:

(b) Compute 
𝑃
(
𝐻
∣
𝐶
)

BlackTom AI · Accepted Answer

Because C is equivalent to H ∪ T (the baby cries iff she is hungry or tired), use conditional probability with set identities.

1) Relevant concepts:
- Conditional probability: P(H|C) = P(H ∩ C) / P(C).
- Equivalence C ↔ (H ∪ T) means C = H ∪ T, so H ∩ C = H.
- Independence: P(H ∩ T) = P(H)P(T) = ht, which is used to compute P(H ∪ T).

2) Step-by-step reasoning:
- Compute P(C) = P(H ∪ T). By the inclusion–exclusion principle and independence:
  P(H ∪ T) = P(H) + P(T) − P(H ∩ T) = h + t − ht.
- Compute P(H ∩ C). Since C = H ∪ T, H ∩ C = H ∩ (H ∪ T) = H. Therefore P(H ∩ C) = P(H) = h.

3) Calculation (all intermediate steps):
- P(C) = h + t − ht.
- P(H ∩ C) = h.
- Thus P(H|C) = P(H ∩ C) / P(C) = h / (h + t − ht).

4) Why this is correct:
- The numerator counts the probability the baby is hungry and crying; because crying occurs exactly when hungry or tired, any hungry state implies crying, so that probability equals P(H)=h. The denominator is the probability the baby cries at all, which is the probability at least one of H or T occurs; independence allows replacing P(H ∩ T) with ht, giving h + t − ht. Dividing yields h/(h + t − ht).

5) Why common incorrect alternatives are wrong:
- h/(h + t) would ignore the overlap (double counting) between H and T.
- h + t − ht (without division) is P(C), not the conditional probability.
- h/(1 − (1 − h)(1 − t)) is algebraically identical to h/(h + t − ht), so it’s acceptable; any form that fails to account for the intersection ht is incorrect.

Conclusion: Therefore P(H|C) = h / (h + t − ht), which matches the provided answer.

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