Advertisements
Advertisements
प्रश्न
A coin is tossed 5 times. What is the probability of getting at least 3 heads?
उत्तर
\[\text{ Let X denote the number of heads in 5 tosses } . \]
\[ \text{ X follows a binomial distribution with n } = 5; p = \text{ probability of getting a head } = \frac{1}{2}\text{ and } q = 1 - p = \frac{1}{2}\]
\[P(X = r) =^{5}{}{C}_r \left( \frac{1}{2} \right)^r \left( \frac{1}{2} \right)^{5 - r} , r = 0, 1, 2 . . . 5\]
\[\text{ The required probability = P(getting at least 3 heads } )\]
\[ = P(X \geq 3) \]
\[ = P(X = 3) + P(X = 4) + P(X = 5)\]
\[ = ^{5}{}{C}_3 \left( \frac{1}{2} \right)^3 \left( \frac{1}{2} \right)^{5 - 3} + ^{5}{}{C}_4 \left( \frac{1}{2} \right)^1 \left( \frac{1}{2} \right)^{5 - 1} + ^{5}{}{C}_5 \left( \frac{1}{2} \right)^5 \left( \frac{1}{2} \right)^{5 - 0} \]
\[= 10 \left( \frac{1}{2} \right)^5 + 5 \left( \frac{1}{2} \right)^5 + 1 \left( \frac{1}{2} \right)^5 \]
\[ = \left( \frac{1}{2} \right)^5 (10 + 5 + 1) \]
\[ = \left( \frac{1}{2} \right)^5 \times 16 \]
\[ = \frac{1}{2}\]
APPEARS IN
संबंधित प्रश्न
There are 6% defective items in a large bulk of items. Find the probability that a sample of 8 items will include not more than one defective item.
A coin is tossed 5 times. What is the probability that tail appears an odd number of times?
A fair coin is tossed 8 times, find the probability of at most six heads.
Find the probability of 4 turning up at least once in two tosses of a fair die.
Bag A contains 1 white, 2 blue and 3 red balls. Bag B contains 3 white, 3 blue and 2 red balls. Bag C contains 2 white, 3 blue and 4 red balls. One bag is selected at random and then two balls are drawn from the selected bag. Find the probability that the balls draw n are white and red.
A committee of 4 persons has to be chosen from 8 boys and 6 girls, consisting of at least one girl. Find the probability that the committee consists of more girls than boys.
A coin and a die are tossed. State sample space of following event.
A: Getting a head and an even number.
A coin and a die are tossed. State sample space of following event.
B: Getting a prime number.
A coin and a die are tossed. State sample space of following event.
C: Getting a tail and perfect square.
Two dice are thrown. Write favourable Outcomes for the following event.
P: Sum of the numbers on two dice is divisible by 3 or 4.
Two dice are thrown. Write favourable outcomes for the following event.
Q: Sum of the numbers on two dice is 7.
Two dice are thrown. Write favourable outcomes for the following event.
R: Sum of the numbers on two dice is a prime number.
Also, check whether Events P and Q are mutually exclusive and exhaustive.
Two dice are thrown. Write favourable outcomes for the following event.
R: Sum of the numbers on two dice is a prime number.
Also, check whether Events Q and R are mutually exclusive and exhaustive.
A card is drawn at random from an ordinary pack of 52 playing cards. State the number of elements in the sample space if consideration of suits is not taken into account.
A card is drawn at random from an ordinary pack of 52 playing cards. State the number of elements in the sample space if consideration of suits is taken into account.
Box-I contains 8 red (R11, R12, R13) and 2 blue (B11, B12) marbles while Box-II contains 2 red(R21, R22) and 4 blue (B21, B22, B23, B24) marbles. A fair coin is tossed. If the coin turns up heads, a marble is chosen from Box-I; if it turns up tails, a marble is chosen from Box-II. Describe the sample space.
Consider an experiment of drawing two cards at random from a bag containing 4 cards marked 5, 6, 7, and 8. Find the sample Space if cards are drawn without replacement.
A car manufacturing factory has two plants, X and Y. Plant X manufactures 70% of cars and plant Y manufactures 30%. 80% of the cars at plant X and 90% of the cars at plant Y are rated of standard quality. A car is chosen at random and is found to be of standard quality. What is the probability that it has come from plant X?
A bag contains 5 red marbles and 3 black marbles. Three marbles are drawn one by one without replacement. What is the probability that at least one of the three marbles drawn be black, if the first marble is red?
Prove that P(A) = `"P"("A" ∩ "B") + "P"("A" ∩ bar"B")`
Bag I contains 3 black and 2 white balls, Bag II contains 2 black and 4 white balls. A bag and a ball is selected at random. Determine the probability of selecting a black ball.
A box has 5 blue and 4 red balls. One ball is drawn at random and not replaced. Its colour is also not noted. Then another ball is drawn at random. What is the probability of second ball being blue?
A die is thrown 5 times. Find the probability that an odd number will come up exactly three times.
An urn contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. Show that the probability of drawing a white ball now does not depend on k.
A die is thrown and a card is selected at random from a deck of 52 playing cards. The probability of getting an even number on the die and a spade card is ______.
A flashlight has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, the probability that both are dead is ______.
Eight coins are tossed together. The probability of getting exactly 3 heads is ______.
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 has failed in mathematics is ______.
A box has 100 pens of which 10 are defective. What is the probability that out of a sample of 5 pens drawn one by one with replacement at most one is defective?
Two cards are drawn together from a pack of 52 cards. The probability that one is a spade and one is a heart, is?
An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red is:
Bag P contains 6 red and 4 blue balls and bag Q contains 5 red and 6 blue balls. A ball is transferred from bag P to bag Q and then a ball is drawn from bag Q. What is the probability that the ball drawn is blue?
Assertion (A): Two coins are tossed simultaneously. The probability of getting two heads, if it is known that at least one head comes up, is `1/3`.
Reason (R): Let E and F be two events with a random experiment, then `P(E/F) = (P(E ∩ F))/(P(E))`.
