Probability and Expected Value: Decision Making Under Risk (Basics) Questions

Question 1

Define probability.

Accepted Answer

Probability is a numerical measure of likelihood of an event, lying between 0 and 1.

Question 2

What is sample space?

Accepted Answer

Sample space is the set of all possible outcomes of a random experiment.

Question 3

State complement rule.

Accepted Answer

P(A') = 1 − P(A).

Question 4

Define independent events.

Accepted Answer

Events are independent if occurrence of one does not affect probability of the other.

Question 5

What is expected value?

Accepted Answer

Expected value is the probability-weighted average of outcomes: EV = Σ(pi×xi).

Question 6

Define conditional probability.

Accepted Answer

Conditional probability P(B|A) is probability of B given A has occurred.

Question 7

A bag has 3 red and 2 blue balls. Find probability of drawing a red ball.

Accepted Answer

Total balls = 3 + 2 = 5. Favourable outcomes (red) = 3.
Probability = 3/5.

Question 8

From a die, find P(even or greater than 4).

Accepted Answer

A = even {2,4,6} so P(A)=3/6=1/2.
B = >4 {5,6} so P(B)=2/6=1/3.
A∩B={6} so P(A∩B)=1/6.
P(A∪B)=1/2 + 1/3 − 1/6 = 2/3.

Question 9

Two coins are tossed. Find probability of getting at least one head.

Accepted Answer

Sample space outcomes: HH, HT, TH, TT (4).
At least one head = {HH, HT, TH} (3 outcomes).
Probability = 3/4.
(Or 1 − P(no head)=1 − 1/4=3/4.)

Question 10

Explain difference between independent and dependent events with one example each.

Accepted Answer

Independent: one event does not change probability of the other. Example: head on first toss and head on second toss (different tosses).
Dependent: probability changes due to earlier event. Example: drawing two cards without replacement (first draw affects second).

Question 11

Compute EMV: High demand p=0.6 payoff 50,000; Low demand p=0.4 payoff 10,000.

Accepted Answer

EMV = 0.6×50,000 + 0.4×10,000 = 30,000 + 4,000 = 34,000.
So expected payoff = ₹34,000.

Question 12

State Bayes theorem formula and explain when it is used (one line).

Accepted Answer

Bayes: P(A|B) = [P(B|A)P(A)]/P(B). It is used to update probability of a cause A after observing evidence B.

Question 13

Explain addition and multiplication rules of probability with examples.

Accepted Answer

Key probability rules:

1) Complement rule:
P(A') = 1 − P(A)

2) Addition rule:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A and B are mutually exclusive, P(A ∩ B)=0 so P(A ∪ B)=P(A)+P(B)

3) Multiplication rule:
If A and B are independent: P(A ∩ B) = P(A)P(B)
If dependent: P(A ∩ B) = P(A)P(B|A)

These rules are used to compute probabilities in business risk situations (e.g., joint events, alternative events).

Question 14

Explain expected value (EMV) method for decision making under risk with an example.

Accepted Answer

Expected Monetary Value (EMV) is the probability-weighted average payoff of a decision.

Steps:
1) List possible states of nature (e.g., high/low demand) with probabilities.
2) List payoffs for each decision under each state.
3) EMV(decision) = Σ(probability × payoff).
4) Choose decision with highest EMV (risk-neutral).

Example:
Prob(high)=0.6, Prob(low)=0.4
Decision 1 payoff: 50,000 (high), 10,000 (low)
Decision 2 payoff: 40,000 (high), 20,000 (low)
EMV1 = 0.6×50,000 + 0.4×10,000 = 34,000
EMV2 = 0.6×40,000 + 0.4×20,000 = 32,000
So Decision 1 is preferred by EMV.

Question 15

Explain conditional probability and Bayes theorem (basic) and mention one business application.

Accepted Answer

Conditional probability is P(B|A) = P(A∩B)/P(A) when P(A)>0.

Bayes’ theorem helps to reverse a conditional probability:
P(A|B) = [P(B|A) × P(A)] / P(B)

Where P(B) can be found using total probability if B can occur due to multiple causes.

In business, Bayes is used in updating probabilities when new information is received (e.g., test results, market signals).

Probability and Expected Value: Decision Making Under Risk (Basics)

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