Advertisements
Advertisements
प्रश्न
A and B take turns in throwing two dice, the first to throw 9 being awarded the prize. Show that their chance of winning are in the ratio 9:8.
उत्तर
\[\text{ Total number of events = 36 } \]
\[P\left( \text{ getting } 9 \right) = \frac{4}{36} = \frac{1}{9}\]
\[P\left( \text{ A winning } \right) = P\left( \text{ getting 9 in first throw } \right) + P\left( \text{ getting 9 in third throw } \right) + . . . \]
\[ = \frac{1}{9} + \left( 1 - \frac{1}{9} \right)\left( 1 - \frac{1}{9} \right) \times \frac{1}{9} + . . . \]
\[ = \frac{1}{9}\left[ 1 + \frac{64}{81} + \left( \frac{64}{81} \right)^2 + . . . \right]\]
\[ = \frac{1}{9}\left[ \frac{1}{1 - \frac{64}{81}} \right] \left[ {1+a+a}^2 {+a}^3 + . . . =\frac{1}{1 - a} \right]\]
\[ = \frac{1}{9} \times \frac{81}{17}\]
\[ = \frac{9}{17}\]
\[P\left( \text{ B winning } \right) = P\left( \text{ getting 9 in second throw } \right) + P\left( \text{ getting 9 in fourth throw } \right) + . . . \]
\[ = \left( 1 - \frac{1}{9} \right)\frac{1}{9} + \left( 1 - \frac{1}{9} \right)\left( 1 - \frac{1}{9} \right)\left( 1 - \frac{1}{9} \right) \times \frac{1}{9} + . . . \]
\[ = \frac{8}{81}\left[ 1 + \frac{64}{81} + \left( \frac{64}{81} \right)^2 + . . . \right]\]
\[ = \frac{8}{81}\left[ \frac{1}{1 - \frac{64}{81}} \right] \left[ {1+a+a}^2 {+a}^3 + . . . =\frac{1}{1 - a} \right]\]
\[ = \frac{8}{81} \times \frac{81}{17}\]
\[ = \frac{8}{17}\]
\[ \therefore \text{ Winning ratio of A to B } = \frac{\frac{9}{17}}{\frac{8}{17}} = \frac{9}{8}\]
APPEARS IN
संबंधित प्रश्न
From a pack of 52 cards, 4 are drawn one by one without replacement. Find the probability that all are aces(or kings).
From a deck of cards, three cards are drawn on by one without replacement. Find the probability that each time it is a card of spade.
Two cards are drawn without replacement from a pack of 52 cards. Find the probability that both are kings .
If A and B are two events such that 2 P (A) = P (B) = \[\frac{5}{13}\] and P (A/B) = \[\frac{2}{5},\] find P (A ∪ B).
The probability that a student selected at random from a class will pass in Mathematics is `4/5`, and the probability that he/she passes in Mathematics and Computer Science is `1/2`. What is the probability that he/she will pass in Computer Science if it is known that he/she has passed in Mathematics?
Assume that each born child is equally likely to be a boy or a girl. If a family has two children, then what is the constitutional probability that both are girls? Given that
(i) the youngest is a girl (b) at least one is a girl.
Given two independent events A and B such that P (A) = 0.3 and P (B) `= 0.6. Find P ( overlineA ∩ B) .`
Given two independent events A and B such that P (A) = 0.3 and P (B) = 0.6. Find P (A/B) .
A die is tossed twice. Find the probability of getting a number greater than 3 on each toss.
Three cards are drawn with replacement from a well shuffled pack of cards. Find the probability that the cards drawn are king, queen and jack.
An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the probability of getting 2 red balls.
An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the probability of getting 2 blue balls.
A bag contains 6 black and 3 white balls. Another bag contains 5 black and 4 white balls. If one ball is drawn from each bag, find the probability that these two balls are of the same colour.
Kamal and Monica appeared for an interview for two vacancies. The probability of Kamal's selection is 1/3 and that of Monika's selection is 1/5. Find the probability that
(i) both of them will be selected
(ii) none of them will be selected
(iii) at least one of them will be selected
(iv) only one of them will be selected.
In a family, the husband tells a lie in 30% cases and the wife in 35% cases. Find the probability that both contradict each other on the same fact.
A husband and wife appear in an interview for two vacancies for the same post. The probability of husband's selection is 1/7 and that of wife's selection is 1/5. What is the probability that none of them will be selected?
An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the sum of the numbers obtained is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered 2, 3, 4, ..., 12 is picked and the number on the card is noted. What is the probability that the noted number is either 7 or 8?
Three digit numbers are formed with the digits 0, 2, 4, 6 and 8. Write the probability of forming a three digit number with the same digits.
Write the probability that a number selected at random from the set of first 100 natural numbers is a cube.
If one ball is drawn at random from each of three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, then the probability that 2 white and 1 black balls will be drawn is
A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is selected randomwise, the probability that it is black or red ball is
An urn contains 9 balls two of which are red, three blue and four black. Three balls are drawn at random. The probability that they are of the same colour is
A bag contains 5 brown and 4 white socks. A man pulls out two socks. The probability that these are of the same colour is
If A and B are two events, then P (`overline A` ∩ B) =
Mark the correct alternative in the following question:
If A and B are two events such that P(A) = \[\frac{4}{5}\] , and \[P\left( A \cap B \right) = \frac{7}{10}\] , then P(B|A) =
Choose the correct alternative in the following question: \[\text{ Let } P\left( A \right) = \frac{7}{13}, P\left( B \right) = \frac{9}{13} \text{ and } P\left( A \cap B \right) = \frac{4}{13} . \text{ Then } , P\left( \overline{ A }|B \right) = \]
Mark the correct alternative in the following question:
\[\text{ If } P\left( B \right) = \frac{3}{5}, P\left( A|B \right) = \frac{1}{2} \text{ and } P\left( A \cup B \right) = \frac{4}{5}, \text{ then } P\left( B|\overline{ A } \right) = \]
Mark the correct alternative in the following question:
\[\text{ If A and B are two independent events with } P\left( A \right) = \frac{3}{5} \text{ and } P\left( B \right) = \frac{4}{9}, \text{ then } P\left( \overline{A} \cap B \right) \text{ equals } \]
Mark the correct alternative in the following question:
\[\text{ If A and B are two independent events such that} P\left( A \right) = 0 . 3 \text{ and } P\left( A \cup B \right) = 0 . 5, \text{ then } P\left( A|B \right) - P\left( B|A \right) = \]
Mark the correct alternative in the following question:
Two dice are thrown. If it is known that the sum of the numbers on the dice was less than 6, then the probability of getting a sum 3, is
An insurance company insured 3000 cyclists, 6000 scooter drivers, and 9000 car drivers. The probability of an accident involving a cyclist, a scooter driver, and a car driver are 0⋅3, 0⋅05 and 0⋅02 respectively. One of the insured persons meets with an accident. What is the probability that he is a cyclist?
Refer to Question 6. Calculate the probability that the defective tube was produced on machine E1.
