From 25 tickets , marked with the first 25 numerals , one is drawn at random. find the chance that it is a multiple of 1) 5 or 7 2)3 or 7
Answers
Answer:
1. Probability
2. Probability It allows us to quantify the variability in the outcome of an experiment whose exact result can’t be predicted with certainty.
3. Definitions Random Experiment A random experiment or trial is one which when conducted successively under the identical conditions, the result is not unique but may be any one of the various possible outcomes. Example: Tossing a fair coin in an experiment.
4. Sample space: The set of all possible outcomes is a sample space. Event: Outcome or combination of outcomes Outcome: The result of an event which we finally achieved is called an outcome or a sample point.
5. Types of Events Mutually exclusive events: Two events are said to be mutually exclusive or incompatible, when both cannot happen simultaneously in a single trial. AÇB=Æ Example: Toss of a coin (either head will occur or tail in a single throw)
6. Independent and dependent events Two or more events are independent when the outcome of one does not affect, and is not affected by the other. Example: if a coin is tossed twice, the result of the second throw would not be affected by the result of the first throw P(AÇB)=P(A).P(B) • Dependent events: occurrence of one event affects probability of happening of other.
7. Equally likely events: events are called equally likely if they have the same chance of occurrence. Example: Throw of unbiased coin (both head and tail have equal chance of occurrence.)
8. Probability Numerical measure (between 0 and 1 inclusively) of the likelihood or chance of occurrence of an uncertain event P(E) = (NO. OF FAVOURABLE OUTCOMES) (NO. OF TOTAL OUTCOMES) 0£ P(E) £ 1
9. Questions for practice A uniform die is thrown. Find the probability that the number on it is (i) Five (ii) greater than 4 (iii)Even no. 2. In a single throw with two uniform dice, find the probability of throwing (i) Both the dice show the same number (ii)A total of Eight (iii) a total of 13 (iv) Total of the numbers on the dice is any number from 2 to 12, both inclusive. 3. A bag contains 4 white, 5 red and 6 green balls.Three balls are drawn at random.What is the chance that a white, a red and a green ball is drawn?
10. 4. Four cards are drawn at random from a pack of 52 cards.Find the probability that They are a king, a queen , a jack or an ace. Two are kings and two are aces. All are diamonds. Two are red and two are black. There are two cards of clubs and two cards of diamonds.
11. 5. Three unbiased coins are tossed.What is the probability of obtaining: All heads Two heads One head At least one head At least two heads All tails
12. 6. Five men in a company of 20 are graduates.If 3 men are picked out of the 20 are random, what is the probability that they all are graduates?What is the probability of at least one graduate? 7. Three groups of workers contain 3 men and one women, 2 men and 2 women, and 1 man and 3 women respectively.One worker is selected at random from each group.What is the probability that the group selected consists of 1 man and 2 women?
13. Answers 1. 1/6, 1/3, ½ 2. 1/6, 5/36, 0,1 3. 24/91 4. 256/52C4, 4C2 x 4C2 / 52C4 , 13C4/ 52C4, 26C2 x 26C2 / 52C4, 13C2 x 13C2 / 52C4. 5. 1/8, 3/8, 3/8,7/8, ½, 1/8 6. 1/114, 137/228 7. 13/32
14. Some Important Results- • 0 £ P(A) £ 1 for all A • P(S) = 1 • P(Ac) = 1 – P(A) for all A • P(A È B) = P(A) + P(B) – P(A Ç B) for all A, B
15. Theorems Of Probability Addition theorem: For two disjoint or mutually exclusive events A&B (i.e P(A Ç B) =0 Since A Ç B=Æ ) P(A È B) = P(A) + P(B) OR P(A OR B)=P(A)+P(B) OR P(A+B)=P(A)+P(B)
16. When events are not mutually exclusive i.e P(A Ç B)≠0 P(A È B) = P(A) + P(B) – P(A Ç B) OR P(A OR B) = P(A) + P(B) – P(A AND B) OR P(A + B) = P(A) + P(B) – P(A.B) For three events A,B & C, P(A È B È C) = P(A)+P(B)+P( C )-P(A ÇB)-P(B ÇC)-P(A ÇC) +P(A ÇB ÇC)
Answer:
p(multiple of 3 or 7) = 5/25 +3/25
=8/25