## 8.11 Set Theoretic

### 8.11.1 Concept

Formulae

Think Why are they so?

- \(P(A\cap B)=P(A)\times P(B)\), if A & B are independent (prove from Bayes theorem)

- \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
- \(P(A\cap \bar B)=P(A)-P(A\cap B)\)
- \(P(A|\bar B)=\frac{P(A \cap \bar B)}{P(\bar B)}=?\)
- Also recall De Morganâ€™s Laws

### 8.11.2 Set Problem # 01

The probability of Ronaldo scoring a goal is 0.4 and that of Messi 0.38. What is the probability that- Both Score
- Only Ronaldo scores
- Only Messi scores?
- At least one of them scores
- Only one of them scores
- At most one of them scores

P(R) = 0.4 and P(M) = 0.38

- \(P(R \cap M)=P(R) \times P(M)\) (since independent)

- \(P(R \cap M')=?\)

- \(P(R' \cap M)=?\)

- \(P(R \cup M)\)

- \(P(R \cap M')+P(R' \cap M)\)

- Same to 5

### 8.11.3 Set Problem # 02

\(S_1\)={1,3,4,7,9,20}

\(S_2\)={12, 13, 14, 15, 16, 17, 18}

If a number is randomly chosen from each set, what is the probability that a prime number comes from \(S_1\) and a multiple of 3 from \(S_2\)?

Say, P = Prime from \(S_1\)

M = Multiple of 3 from \(S_2\).

\(P(P) = \frac 3 7\) and \(P(M)=\frac 3 7\)

What do we need to find out?

- \(P(P \cup M)\)
- \(P(P \cap M)\)

- Answer: \(P(P \cap M) = P(P) \cdot P(M)\)

### 8.11.4 Set Problem # 03

Cup 01 contains 2 black, 3 red, and 1 pink ball. Cup 2 contains only 1 red ball. A cup is selected randomly. Next, a ball is randomly chosen from that randomly selected cup and placed into the other cup. A ball is then drawn randomly from that second cup. Find the probability that the last ball drawn is a pink one.

- 3 possible cases

- 1st cup \(\rightarrow\) pink ball \(\rightarrow\) pink ball from 2nd cup

- 1st cup \(\rightarrow\) non-pink ball \(\rightarrow\) pink ball from 2nd cup

- 2nd cup \(\rightarrow\) red ball \(\rightarrow\) pink ball from 1st (other) cup

- \((\frac 1 2 \times \frac 1 6 \times \frac 1 2 )+ (\frac 1 2 \times \frac 5 6 \times 0) + (\frac 1 2 \times 1 \times \frac 1 7)\)

### 8.11.5 Set problem # 04

If a senility researcher discovered that in a population of healthy and diseased elderly people, 14% of the people had senile dementia, 63% had arterioplerotic cerebral degeneration, and 11% had both. What is the probability that a person not having arterioplerotic cerebral degeneration has senile dementia?

### 8.11.6 Set problem # 05

A candidate applied for three posts in an industry, having 3, 4, and 2 candidates respectively. What is the probability of getting a job by that candidate in at least one post?

\(P(F)+P(S)+P(T)-P(F\cap S)-P(S\cap T)-P(F\cap T)+P(F\cap S \cap T\)

### 8.11.7 Set problem # 06

A card is drawn from each of two well-shuffled pack of cards. Find the probability that at least one of them is an ace.

Let,

A = Ace from 1st pack

B = Ace from 2nd pack

\(P(A \cup B)=?\)

\(P(A) = \frac{^4C_1}{^{52}C_1}\)

\(P(B) = P(A)\)

\(P(A\cap B) = P(A) \times P(B)\)

- \(P(A \cup B)=P(A)+P(B) - P(A \cap B)\)

### 8.11.8 Set problem # 07

\(P(A\cap B)= \frac 1 3, P(A \cup B) = \frac 5 6, and \space P(A) = \frac 1 2\)

Find \(P(B)\) and \(P(B^c)\)

### 8.11.9 Set problem # 08

\(P(A)= \frac 1 2, P(B) = \frac 1 5, \text{and} \space P(A|B) = \frac 3 8\)

Find \(P(A \cap B), P(B|A)\), and \(P(A \cup B)\)

\(P(B|A) = \frac {P(A \cap B)}{P(A)}\)

\(P(A|B) = \frac {P(A \cap B)}{P(B)}\)

\(P(A \cap B) = P(A|B) \times P(B)\)

\(\therefore P(B|A) = \frac{P(A|B) \times P(B)}{P(A)}\)