Advertisements
Advertisements
प्रश्न
Find the probability of throwing at most 2 sixes in 6 throws of a single die.
उत्तर
Let X denote the number of sixes.
P (getting a six when a die is thrown) = p = `(1)/(6)`
∴ q = 1 – p = `1 - (1)/(6) = (5)/(6)`
Given, n = 6
∴ X ∼ B`(6, 1/6)`
The P.M.F. of X is given by
P(X = x) = `""^6"C"_x (1/6)^x (5/6)^(6 - x), x` = 0, 1, ..., 6
P (getting at most 2 sixes)
= P(X ≤ 2)
= P(X = 0 or X = 1 or X = 2)
= P(X = 0) + P(X = 1) + P(X = 2)
= `""^6"C"_0(1/6)^0 (5/6)^(6 - 0) + ""^6"C"_1(1/6)^1(5/6)^(6 - 1) + ""^6"C"_2(1/6)^2 (5/6)^(6 - 2)`
= `1 xx 1 xx (5/6)^6 + 6 xx 1/6 (5/6)^5 + (6 xx 5)/(2 xx 1) xx 1/6 xx 1/6 xx (5/6)^4`
= `(5/6)^4 [(5/6)^2 + 5/6 + 5/12]`
= `(5/6)^4 [25/36 + 30/36 + 15/36]`
P(X ≤ 2) = `(5/6)^4[(25 + 30 +15)/36]`
P(X ≤ 2) = `(5/6)^4 (70/36)`
P(X ≤ 2) = `(5/6)^4 (35/18)`
P(X ≤ 2) = `(5/6)^4 ((5 xx 7)/(6 xx 3))`
P(X ≤ 2) = `(7/3)(5/6)^5`
APPEARS IN
संबंधित प्रश्न
Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is hostler?
Find the probability distribution of number of tails in the simultaneous tosses of three coins.
Find the probability distribution of number of heads in four tosses of a coin.
A random variable X has the following probability distribution.
| X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| P(X) | 0 | k | 2k | 2k | 3k | k2 |
2k2 |
7k2 + k |
Determine
(i) k
(ii) P (X < 3)
(iii) P (X > 6)
(iv) P (0 < X < 3)
Find the probability distribution of the number of doublets in four throws of a pair of dice. Also find the mean and variance of this distribution.
Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
Find the probability distribution of Y in two throws of two dice, where Y represents the number of times a total of 9 appears.
Three cards are drawn successively with replacement from a well-shuffled deck of 52 cards. A random variable X denotes the number of hearts in the three cards drawn. Determine the probability distribution of X.
A fair coin is tossed four times. Let X denote the longest string of heads occurring. Find the probability distribution, mean and variance of X.
In roulette, Figure, the wheel has 13 numbers 0, 1, 2, ...., 12 marked on equally spaced slots. A player sets Rs 10 on a given number. He receives Rs 100 from the organiser of the game if the ball comes to rest in this slot; otherwise he gets nothing. If X denotes the player's net gain/loss, find E (X).
If the probability distribution of a random variable X is as given below:
Write the value of P (X ≤ 2).
| X = xi : | 1 | 2 | 3 | 4 |
| P (X = xi) : | c | 2c | 4c | 4c |
Mark the correct alternative in the following question:
For the following probability distribution:
| X: | −4 | −3 | −2 | −1 | 0 |
| P(X): | 0.1 | 0.2 | 0.3 | 0.2 | 0.2 |
The value of E(X) is
Three fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X.
For the following probability density function (p. d. f) of X, find P(X < 1) and P(|x| < 1)
`f(x) = x^2/18, -3 < x < 3`
= 0, otherwise
Write the negation of the following statements :
(a) Chetan has black hair and blue eyes.
(b) ∃ x ∈ R such that x2 + 3 > 0.
If X ∼ N (4,25), then find P(x ≤ 4)
The p.m.f. of a random variable X is
`"P"(x) = 1/5` , for x = I, 2, 3, 4, 5
= 0 , otherwise.
Find E(X).
Determine whether each of the following is a probability distribution. Give reasons for your answer.
| z | 3 | 2 | 1 | 0 | -1 |
| P(z) | 0.3 | 0.2 | 0.4. | 0.05 | 0.05 |
Find the probability distribution of the number of successes in two tosses of a die if success is defined as getting a number greater than 4.
A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of 2 successes
A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at least 3 successes
State whether the following is True or False :
If r.v. X assumes the values 1, 2, 3, ……. 9 with equal probabilities, E(x) = 5.
Solve the following problem :
Following is the probability distribution of a r.v.X.
| x | – 3 | – 2 | –1 | 0 | 1 | 2 | 3 |
| P(X = x) | 0.05 | 0.1 | 0.15 | 0.20 | 0.25 | 0.15 | 0.1 |
Find the probability that X is non-negative
Solve the following problem :
Find the probability of the number of successes in two tosses of a die, where success is defined as number greater than 4.
Solve the following problem :
Find the probability of the number of successes in two tosses of a die, where success is defined as six appears in at least one toss.
Solve the following problem :
The probability that a lamp in the classroom will burn is 0.3. 3 lamps are fitted in the classroom. The classroom is unusable if the number of lamps burning in it is less than 2. Find the probability that the classroom cannot be used on a random occasion.
Solve the following problem :
The probability that a component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 components tested survive.
Solve the following problem :
A computer installation has 3 terminals. The probability that any one terminal requires attention during a week is 0.1, independent of other terminals. Find the probabilities that 1 terminal requires attention during a week.
Find the mean and variance of the number randomly selected from 1 to 15
Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean or expectation of X and variance of X
The probability distribution of a discrete random variable X is given as under:
| X | 1 | 2 | 4 | 2A | 3A | 5A |
| P(X) | `1/2` | `1/5` | `3/25` | `1/10` | `1/25` | `1/25` |
Calculate: The value of A if E(X) = 2.94
The probability distribution of a random variable x is given as under:
P(X = x) = `{{:("k"x^2, "for" x = 1"," 2"," 3),(2"k"x, "for" x = 4"," 5"," 6),(0, "otherwise"):}`
where k is a constant. Calculate P(X ≥ 4)
For the following probability distribution:
| X | 1 | 2 | 3 | 4 |
| P(X) | `1/10` | `3/10` | `3/10` | `2/5` |
E(X2) is equal to ______.
A bag contains 1 red and 3 white balls. Find the probability distribution of the number of red balls if 2 balls are drawn at random from the bag one-by-one without replacement.
Two numbers are selected from first six even natural numbers at random without replacement. If X denotes the greater of two numbers selected, find the probability distribution of X.
The probability that a bomb will hit the target is 0.8. Complete the following activity to find, the probability that, out of 5 bombs exactly 2 will miss the target.
Solution: Here, n = 5, X =number of bombs that hit the target
p = probability that bomb will hit the target = `square`
∴ q = 1 - p = `square`
Here, `X∼B(5,4/5)`
∴ P(X = x) = `""^"n""C"_x"P"^x"q"^("n" - x) = square`
P[Exactly 2 bombs will miss the target] = P[Exactly 3 bombs will hit the target]
= P(X = 3)
=`""^5"C"_3(4/5)^3(1/5)^2=10(4/5)^3(1/5)^2`
∴ P(X = 3) = `square`
A large chain retailer purchases an electric device from the manufacturer. The manufacturer indicates that the defective rate of the device is 10%. The inspector of the retailer randomly selects 4 items from a shipment. Complete the following activity to find the probability that the inspector finds at most one defective item in the 4 selected items.
Solution:
Here, n = 4
p = probability of defective device = 10% = `10/100 = square`
∴ q = 1 - p = 1 - 0.1 = `square`
X ∼ B(4, 0.1)
`P(X=x)=""^n"C"_x p^x q^(n-x)= ""^4"C"_x (0.1)^x (0.9)^(4 - x)`
P[At most one defective device] = P[X ≤ 1]
= P[X=0] + P[X=1]
= `square+square`
∴ P[X ≤ 1] = `square`
Five numbers x1, x2, x3, x4, x5 are randomly selected from the numbers 1, 2, 3, ......., 18 and are arranged in the increasing order such that x1 < x2 < x3 < x4 < x5. What is the probability that x2 = 7 and x4 = 11?
