probability

Probability is a branch of mathematics that deals with calculating the likelihood of given event’s occurrence, which is expressed as a number between 0 and 1. We usually predict about many events based on certain parameters.

E.g.: 1. Getting a head , when a coin is tossed.

2. Getting a odd number, when a die is rolled. The better we know about the parameters related to an event result then the result predicted accuracy is better.
 

Model Problems 

1. When three coins are tossed simultaneously, what is the probability that one coins show tails as output? 

Explanation: total outcomes = 2³ = 8 Favourable outcomes (3 coins we can select one tail) = 3C₁ = 3 Probability = 3/8

Ans: B 

2. When five coins are tossed simultaneously, what is the probability that at least three coins show tails as output? 

Explanation: total outcomes = 2⁵ = 32 Favourable outcomes (5 coins we can select atleast three tails) = 5C₃ + 5C₄ + 5C₅ = 3 =10 + 5 + 1 = 16 

Ans: C

3. What is the probability of having 53 tuesdays in a non-leap year?

Explanation: In a non-leap year, there are 365 days weekly 7 days so no. of weeks 

52 weeks and 1 day is left that day should be tuesday. The total outcomes of that 1 day is = {sun, mon, tue, wed, thurs, fri, sat} = 7. Favourable outcomes of that 1 day is tuesday = {tue} = 1 

Probability = 1/7

 Ans: A

4. What is the probability of having 53 wednesdays in a leap year?

Explanation: In a leap year, there are 366 days weekly 7 days so no. of weeks ]

52 weeks and 2 days are left in that one day should be wednesday. The total outcomes of that 2 days are = {(sun, mon), (mon, tue), (tue, wed), (wed, thurs), (thurs, fri), (fri, sat) (sat, sun)} = 7. Favourable outcomes of that 1 day is wednesday in that 2 days = {(tues, wed) (wed, thurs)} = 2

Probability = 2/7

 Ans: B

5. A single letter is selected at random from the word "KIRAN". The probability that it is a vowel, is? 

Explanation: total outcomes (total no. of letters) = 5 favourable outcomes (total no. of vowels )= 2  probability = 2/5

Ans: B

6. When an unbiased dice is rolled, what is the probability that the output is greater than 4? 

Explanation: total outcomes = {1, 2, 3, 4, 5, 6} = 6 favourable outcomes (greater than 4) = {5, 6} = 2 

Ans: B

7. When two dice are rolled together, what is the probability that the sum of the outputs is 9?  

Ans: D

8. When two dice are rolled together, what is the probability that the sum of the outputs is 7 or 12? 

Explanation: Probability that the sum of the outputs is 

probability that the sum of the outputs is 

probability that the sum of the outputs is 7 or 12 

Ans: B

Directions: A basket contains 4 red, 5 blue and 3 green marbles.

9. If two marbles are drawn at random, what is the probability that both are red? 

Explanation: total outcomes 

favourable outcomes = 

clubs(13)  spades(13)   hearts(13)    diamonds(13) 

In every suit of clubs or spades or hearts or diamonds ace, 2, 3, 4, 5, 6, 7, 8, 9,10, jack, queen, king (13 cards).

 There are 13 cards of each suit clubs or spades or hearts or diamonds. 

There are 4 ace, 4 jacks, 4 queeens, 4 kings. 

There are 12 face cards (king, queen, jack are face cards) 

There are 16 honour cards (ace, king, queen, jack are honour cards)
 

Examples:

A single card is chosen at a random from standard deck of playing 52 cards.

i. What is the probability that choosing a king?

Sol: Total outcomes (total playing cards) = 52C₁ = 52 (one card is selected from total cards) Favourable outcomes (total kings) = 4C₁ = 4 (one card is selected from total kings) 

 

ii. What is the probability that choosing a black card?

Sol: Total outcomes (total playing cards) = 52C₁ = 52 (one card is selected from total cards) Favourable outcomes (total black cards) =  13C₁ = 13 (one card is selected from total black cards) 

iii. What is the probability that choosing a face card?

Sol: Total outcomes (total playing cards) = 52C₁ = 52 (one card is selected from total cards) Favourable outcomes(total face cards) = 12C₁ = 12

(one card is selected from total face cards) 

Model problems

A card is drawn at random from a pack of 52 cards what is the probability of getting... 

1. A spade card? 

Explanation: Total number of outcomes = 52C₁ = 52 Favourable no.of outcomes =13C₁ = 13 

Required probability = 

Ans: B
2. A heart card? 

Explanation: Total number of outcomes = 52C₁ = 52 Favourable no.of outcomes =13C₁ = 13 

Required probability

Ans: B

3. Jack card?

Explanation: Total number of outcomes = 52C₁ = 52 Favourable no.of outcomes = 4C₁ = 4 

Required probability 

Ans:C

4. A red face card? 

Explanation: Total number of outcomes = 52C₁ = 52 Favourable no.of outcomes = 6C₁ = 6 

Ans:A