Advertisements
Advertisements
Question
Mark the correct alternative in the following question:
\[ \text{ If } P\left( B \right) = \frac{3}{5}, P\left( A|B \right) = \frac{1}{2} \text{ and } P\left( \overline{A \cup B }\right) = \frac{4}{5}, \text{ then } P\left( \overline{ A } \cup B \right) + P\left( A \cup B \right) = \]
Options
\[\frac{1}{5}\]
\[ \frac{4}{5} \]
\[ \frac{1}{2} \]
\[ 1\]
Solution
\[\text{ We have } , \]
\[P\left( B \right) = \frac{3}{5}, P\left( A|B \right) = \frac{1}{2} \text{ and } P\left( A \cup B \right) = \frac{4}{5}\]
\[\text{ As } , P\left( A|B \right) = \frac{1}{2}\]
\[ \Rightarrow \frac{P\left( A \cap B \right)}{P\left( B \right)} = \frac{1}{2}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{1}{2} \times P\left( B \right)\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{1}{2} \times \frac{3}{5}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{3}{10}\]
\[\text{ As } , P\left( A \cup B \right) = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) + P\left( B \right) - P\left( A \cap B \right) = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) + \frac{3}{5} - \frac{3}{10} = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) + \frac{3}{10} = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) = \frac{4}{5} - \frac{3}{10}\]
\[ \Rightarrow P\left( A \right) = \frac{5}{10}\]
\[ \Rightarrow P\left( A \right) = \frac{1}{2}\]
\[\text{ Now } , \]
\[P\left(\overline{ A \cup B } \right) + P\left( \overline{ A } \cup B \right) = \left[ 1 - P\left( A \cup B \right) \right] + \left[ 1 - P\left( \text {Only A } \right) \right]\]
\[ = \left[ 1 - \frac{4}{5} \right] + 1 - \left[ P\left( A \right) - P\left( A \cap B \right) \right]\]
\[ = \frac{1}{5} + 1 - \left[ \frac{1}{2} - \frac{3}{10} \right]\]
\[ = \frac{6}{5} - \frac{2}{10}\]
\[ = \frac{10}{10}\]
\[ = 1\]
APPEARS IN
RELATED QUESTIONS
In a set of 10 coins, 2 coins are with heads on both the sides. A coin is selected at random from this set and tossed five times. If all the five times, the result was heads, find the probability that the selected coin had heads on both the sides.
How many times must a fair coin be tossed so that the probability of getting at least one head is more than 80%?
An experiment succeeds thrice as often as it fails. Find the probability that in the next five trials, there will be at least 3 successes.
A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of
(i) 5 successes?
(ii) at least 5 successes?
(iii) at most 5 successes?
Ten cards numbered 1 through 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number?
Two cards are drawn without replacement from a pack of 52 cards. Find the probability that both are kings .
Two cards are drawn without replacement from a pack of 52 cards. Find the probability that the first is a king and the second is an ace.
If A and B are two events such that P (A ∩ B) = 0.32 and P (B) = 0.5, find P (A/B).
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).
A coin is tossed three times. Find P (A/B) in each of the following:
A = At most two tails, B = At least one tail.
A card is drawn from a pack of 52 cards so the teach card is equally likely to be selected. In which of the following cases are the events A and B independent?
A = the card drawn is black, B = the card drawn is a king.
Given two independent events A and B such that P (A) = 0.3 and P (B) = 0.6. Find \[P \overline A \cup \overline B \] .
If P (not B) = 0.65, P (A ∪ B) = 0.85, and A and B are independent events, then find P (A).
If A and B are two independent events such that P (`bar A` ∩ B) = 2/15 and P (A ∩`bar B` ) = 1/6, then find P (B).
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.
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that both the balls are red.
A bag contains 8 red and 6 green balls. Three balls are drawn one after another without replacement. Find the probability that at least two balls drawn are green.
Arun and Tarun appeared for an interview for two vacancies. The probability of Arun's selection is 1/4 and that to Tarun's rejection is 2/3. Find the probability that at least one of them will be selected.
A and B toss a coin alternately till one of them gets a head and wins the game. If A starts the game, find the probability that B will win the game.
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 both of them will be selected ?
X is taking up subjects - Mathematics, Physics and Chemistry in the examination. His probabilities of getting grade A in these subjects are 0.2, 0.3 and 0.5 respectively. Find the probability that he gets
(i) Grade A in all subjects
(ii) Grade A in no subject
(iii) Grade A in two subjects.
A bag contains 8 marbles of which 3 are blue and 5 are red. One marble is drawn at random, its colour is noted and the marble is replaced in the bag. A marble is again drawn from the bag and its colour is noted. Find the probability that the marble will be
(i) blue followed by red.
(ii) blue and red in any order.
(iii) of the same colour.
The bag A contains 8 white and 7 black balls while the bag B contains 5 white and 4 black balls. One ball is randomly picked up from the bag A and mixed up with the balls in bag B. Then a ball is randomly drawn out from it. Find the probability that ball drawn is white.
A four digit number is formed using the digits 1, 2, 3, 5 with no repetitions. Write the probability that the number is divisible by 5.
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.
A ordinary cube has four plane faces, one face marked 2 and another face marked 3, find the probability of getting a total of 7 in 5 throws.
If A and B are two independent events, then write P (A ∩ \[B\] ) in terms of P (A) and P (B).
If A and B are independent events, then write expression for P(exactly one of A, B occurs).
A and B draw two cards each, one after another, from a pack of well-shuffled pack of 52 cards. The probability that all the four cards drawn are of the same suit is
Choose the correct alternative in the following question:
\[\text{ If } P\left( A \right) = \frac{2}{5}, P\left( B \right) = \frac{3}{10} \text{ and } P\left( A \cap B \right) = \frac{1}{5}, \text{ then } , P\left( \overline { A }|\overline{ B } \right) P\left( \overline{ B }|\overline{ A } \right) \text{ is equal to } \]
Mark the correct alternative in the following question:
\[\text{ If A and B are two events such that } P\left( A \right) = \frac{1}{2}, P\left( B \right) = \frac{1}{3}, P\left( A|B \right) = \frac{1}{4}, \text{ then } P\left( A \cap B \right) \text{ equals} \]
If A and B are two events such that A ≠ Φ, B = Φ, then
Mark the correct alternative in the following question:
In a college 30% students fail in Physics, 25% fail in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in Physics if she failed in Mathematics is
Mark the correct alternative in the following question
Three persons, A, B and C fire a target in turn starting with A. Their probabilities of hitting the target are 0.4, 0.2 and 0.2, respectively. The probability of two hits is
Mark the correct alternative in the following question:
If two events are independent, then
Mark the correct alternative in the following question:
\[\text{ If A and B are two events such that } P\left( A|B \right) = p, P\left( A \right) = p, P\left( B \right) = \frac{1}{3} \text{ and } P\left( A \cup B \right) = \frac{5}{9}, \text{ then} p = \]
