This document defines key concepts in probability, including: - Probability is how likely something is to happen, with all probabilities between 0 and 1. - Events have probabilities calculated by the number of outcomes of the event divided by the total number of outcomes. - Conditional probability is the probability of one event given that another event occurred, calculated as the probability of both events occurring divided by the probability of the first event. - Independent events do not affect each other's probabilities, while dependent events do influence each other's probabilities.

1. Probability
561 – Sachinkumar Patel
562 – Prakash Prasad
663 – Shreya Rana
564 – Pratham Raut
565 – Srushti Redkar

2. Probability
 How likely something is to happen.
 All probabilities are between 0 and 1.
 Event
 Simple Event
 Sample Space
 Formula =
 P(A) = n(A)/n(S)

3. History of probability
 17th century
 By Pierre de Fermat and Blaise Pascal.

4. Conditional Probability
 If A and B are two events associated with the same sample space of
a random experiment, the conditional probability of event A given
that B has occurred is given by p(A/B)=P( A ∩ B ) / P( B ), provided
P( B ) ≠ 0.
 Formula =
 P(A | B) = P(A ∩ B) / P(B) ( P(B) ≠ 0 )
 P(B | A) = P(A ∩ B) / P(A) ( P(A) ≠ 0 )

5. Independent Events and Dependent Events
 Independent Event
- The outcome of one event does
not affect the outcome of the
other event.
- Formula =
- P( A ∩ B ) = P( A ) * P( B )
 Dependent Event
- The outcome of one event does
influence the outcome of the other
event.
- Formula =
- P( A ∩ B ) = P( A ) * P( B after A )

6. Mutually exclusive events
 The events are mutually exclusive if they don’t occur same time or
simultaneously.
 Formula =
 P (A ∪ B) = P(A) + P(B)

7. Probability Distribution
 Random Variable = X
- A type of variable in statistics whose possible values depend on the
outcomes of a certain random phenomenon.
• Probability Distribution = P(X)
- If a random variable x takes values x1, x2, …., xn with respective
probability p1, p2, …., pn, then is known as the probability distribution of
X.
 Formula =
 𝑖=1
𝑛
𝑝𝑖 = 1
X X1 X2 …….. Xn
P(X) P1 P2 …….. Pn

8. Sums 561

9. R Implementation 561

10. Sum 562

11. R Implementation 562

12. Sum 563

13. R Implementation 563

14. Sum 564

15. R Implementation 564

16. Sum 565

17. R Implementation 565

18. Application of probability
 Flipping a coin
 Choosing a card from the deck
 Throwing a dice
 Pulling a green candy from a bag of red candies
 Winning a lottery 1 in many millions
 Weather forecasting
 Calculation of batting average in cricket

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