Important concepts

• Trial
• Experiment
• Random experiment
• Sample space
• event

Classical

$P (A) = \frac{n(A)}{n(S)}$

Relative frequency

$$\lim_{n(S) \to \infty} \frac{n(A)}{n(S)}$$

Axiomatic

Three axioms

Say, S is sample space and A is an event

• $0 \le P (A) \le 1$ (NOT $P(A) \ge 0$)
• At least one of S will occur. P (S) = 1; Certain event.
• $P(A_1 U A_2 U ... U A_n)=P(A_1) + P(A_2) + ... + P(A_n)$ or
  
• $$P\left(\cup _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }P(E_{i})$$

Permutaion vs Combination

There are 15 cricketers in BD preliminary team. We got to select 11. C or P?

• Where is P used?

Types of Problems

• Miscellaneous
  
• Coin
• Die
• Playing Card
• At Once vs One by One
• Box
• Conditional
• Set Theoretic
• Digit

Misc Problem #01

What is the probability that in a leap year, there are 53 Fridays?

• In a leap year, there are 366 days, i.e, 52 weeks and 2 days. In each week is a Fridays, so there are no less than 52 Fridays. The remaining two days could be:
• (Sat, Sun); (Sun, Mon); (Mon, Tue); (Tue, Wedn); (Wedn, Thu); (Thu, Fri); (Fri, Sat) = 7
• Total possible outcome = 7 and favorable outcomes = 2
• $P = \frac{2}{7}$

Misc Problem #02

Out of the natural numbers 10 through 30, a number is chosen randomly; what is the probability that the number is

1. a prime number
2. a prime number or multiple of 5
3. a prime number or an odd number
4. not a perfect square

Misc Problem #03

What is the probability that the product of three positive integers chosen from 1 through 100 is an even number?

• Three possible cases
• All three are even
• Two odd and one even number
• Two even and one odd
• $P=\frac{^{50}C_3}{^{100}C_3}+...$
• 0.88

Tossing A Coing Twice

Tossing a coin twice is equivalent to tossing two coins at once

Flipping A Coin Thrice

What is the probability that

Flinging Two Dice at Once

What is the probability that

Concepts (Playing Card)

Card Problem #01

3 cards are drawn from a pack of 52 cards. What is the probability that they are all Kings?

There are 4 Kings. We’ve to draw 3 cards.

• $P(K) =\frac{^4C_3}{{^52}C_3}$

Card Problem #02

If a card is drawn from a deck of 52 cards with 4 aces, what is the probability that an ace will not show up?

Let, P(A) = Ace appears

• $1-P(A)$
• $1-\frac 1 {13}$

Card Problem #03

Two cards are drawn with replacement; What is the probability that they are

1. Kings of same color
2. Kings of different color
3. Not Kings at all

• 1. $P(BUR) =P(B)+P(R)$
• $\frac{^2C_1 \times ^2C_1}{^{52}C_1\times ^{52}C_1}+\frac{^2C_1 \times ^2C_1}{^{52}C_1\times ^{52}C_1}$ Why not $^{52}C_2$, $^4C_2$

• 2. $1-P(B \cup R)$
• $P(K)= \frac{^4C_1 \times ^4C_1}{^{52}C_1\times ^{52}C_1}$
• 3. $1-P(K)$

Card Problem #04

A card is drawn from a pack of 52 cards. What is the probability that it is

1. an Ace
2. A Spade
3. A Hearts or a King

Card Problem #05

Box Problem #01

In a box, there are 5 blue marbles, 7 green marbles, and 8 yellow marbles. If two marbles are randomly selected, what is the probability that both will be green or yellow, if taken

• Correct or not: $\frac{^7C_2}{^{20}C_2}+\frac{^8C_2}{^{20}C_2}$
• $\frac{^7C_1 \times ^7C_1}{^{20}C_1 \times ^{20}C_1}+\frac{^8C_1 \times ^8C_1}{^{20}C_1 \times ^{20}C_1}$
• Without replacement: $\frac{^7C_1 \times ^6C_1}{^{20}C_1 \times ^{20}C_1}+\frac{^8C_1 \times ^7C_1}{^{20}C_1 \times ^{20}C_1}$

Box Problem #02

Box Problem #02

There are 9 red and 7 white balls in a box. 6 balls are picked randomly. What is the probability that 3 balls are red and 3 balls are white?

Which one is the answer?

Conditional Formula

Bayes Theorem

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

Conditional Problem # 01

Probability that it rains today is 40%, that tomorrow is 50%, and that on both days is 30%. If it rains today, what is the probability that it would rain tomorrow?

• $P (T) = 0.4, P(M) = 0.5, P(T\cap M)=0.3$
• $P(M|T)=?$
• $P(M|T)=\frac{P(T\cap M)}{P(T)}$
• $\frac{0.3}{0.4}$

Conditional Problem # 02

In a college, there are 100 students, of whom 30 play football, 40 play cricket, and 20 play both. A student is selected randomly. If he plays cricket, what is the probability that he plays football?

Conditional Problem # 03

$S=$ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

If a number is picked randomly and known it an even number, What is the probability that it is more than 6?

Conditional Problem # 04

In a city of 1 million inhabitants let there be 100 terrorists and 999,900 non-terrorists. The city has a facial recognition software. If the camera scans a terrorist, a bell will ring 99% of the time, and it will fail to ring 1% of the time. If the camera scans a non-terrorist, a bell will not ring 99% of the time, but it will ring 1% of the time.

Concept

Formulae

Set Problem # 01

Set Problem # 02

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

1. • 1st cup $\rightarrow$ pink ball $\rightarrow$ pink ball from 2nd cup
2. • 1st cup $\rightarrow$ non-pink ball $\rightarrow$ pink ball from 2nd cup
3. • 2nd cup $\rightarrow$ red ball $\rightarrow$ pink ball from 1st (other) cup

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

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?

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?

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.

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 $P(B^c)$

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 $P(A \cup B)$