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:

(a) Compute (P(C)) in terms of (h) and (t).

BlackTom AI · Accepted Answer

Because crying occurs exactly when H or T (or both) occur, P(C) equals the probability of the union H ∪ T.

1. Relevant concepts
- Probability of a union: P(H ∪ T) = P(H) + P(T) − P(H ∩ T).
- Independence: If H and T are independent, P(H ∩ T) = P(H)P(T).
- Complement rule (alternative): P(H ∪ T) = 1 − P((H ∪ T)^c) = 1 − P(H^c ∩ T^c).

2. Step-by-step reasoning
- Since C ↔ (H ∪ T), P(C) = P(H ∪ T).
- Apply the union formula: P(H ∪ T) = P(H) + P(T) − P(H ∩ T).

3. Calculations (all intermediate steps)
- Given P(H) = h and P(T) = t, and independence implies P(H ∩ T) = h·t.
- Substitute: P(C) = h + t − (h·t) = h + t − ht.
- Alternative check via complement: P(C) = 1 − P(H^c ∩ T^c) = 1 − (1 − h)(1 − t) = 1 − (1 − h − t + ht) = h + t − ht. Same result.

4. Why this is correct
- The union formula corrects for double-counting outcomes where both H and T occur; independence gives the exact product ht for that overlap. Subtracting ht removes the overlap counted twice, yielding the correct probability of at least one condition (hungry or tired) being true.

5. Why other simple expressions would be wrong
- P(C) = h or = t would ignore the other cause entirely. P(C) = ht would give only the joint occurrence, not the cases where exactly one condition holds. P(C) = h + t (without subtracting ht) can exceed 1 and double-counts the both-occurring case. Thus h + t − ht is the only correct expression.

6. Conclusion
The probability that the baby is crying is P(C) = h + t − ht, which matches the provided answer.

Similar Questions

The probability of Event A occurring is 0.3 and the probability of Event B occurring is 0.5. If Event A and Event B are independent, then what is the joint probability of Event A and Event B both occurring?

Chai walks to college 60% of the time. Vy does the same 30% of the time. Considering these events as independent, the probability that both students walk to college is:

An experiment consists of tossing a coin and then rolling a fair six-sided die. What isthe probability of observing a head and a ‘six’?

The product rule is used to solve problems in which

If p=0.11 and q=0.62, what is the probability that both A and B happen?

Are E1 and E2 independent events?

10 If a penny is tossed four times and comes up heads all four times, the probability of heads on the fifth trial is

Suppose that two fair dice are tossed. Let Ai be the event that a 6 shows on the ith die, i=1,2. Then P(A1∩A2) is:

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler