SATHEE: Probability 1 Question 17

Probability 1 Question 17

25. An unbiased die, with faces numbered 1, 2, 3, 4, 5 and 6 is thrown $n$ times and the list of $n$ numbers showing up is noted. What is the probability that among the numbers 1, 2, 3, 4, 5 and 6 only three numbers appear in this list?

(2001, 5M)

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Answer:

Correct Answer: 25. $\frac{(3^{n} - 3 \cdot 2^{n} + 3) \times^{6} C_{3}}{6^{n}}$

Solution:

Let us define a onto function $F$ from $A : [r_{1}, r_{2}, \dots, r_{n}]$ to $B : [1, 2, 3]$ , where $r_{1}, r_{2}, \dots, r_{n}$ are the readings of $n$ throws and 1,2,3 are the numbers that appear in the $n$ throws.

Number of such functions, $M = N - [n (1) - n (2) + n (3)]$ where, $N =$ total number of functions and $n (t) =$ number of function having exactly $t$ elements in the range.

$\begin{array}{ll} Now, & N = 3^{n}, n (1) = 3 \cdot 2^{n}, n (2) = 3, n (3) = 0 \\ \Rightarrow & M = 3^{n} - 3 \cdot 2^{n} + 3 \end{array}$

Hence, the total number of favourable cases

$\begin{aligned} = (3^{n} - 3 \cdot 2^{n} + 3) \cdot^{6} C_{3} \\ ∴ Required probability & = \frac{(3^{n} - 3 \cdot 2^{n} + 3) \times^{6} C_{3}}{6^{n}} \end{aligned}$

Probability 1 Question 17

25. An unbiased die, with faces numbered 1, 2, 3, 4, 5 and 6 is thrown $n$ times and the list of $n$ numbers showing up is noted. What is the probability that among the numbers 1, 2, 3, 4, 5 and 6 only three numbers appear in this list?

Answer:

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Probability 1 Question 17

25. An unbiased die, with faces numbered 1, 2, 3, 4, 5 and 6 is thrown n times and the list of n numbers showing up is noted. What is the probability that among the numbers 1, 2, 3, 4, 5 and 6 only three numbers appear in this list?

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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25. An unbiased die, with faces numbered 1, 2, 3, 4, 5 and 6 is thrown $n$ times and the list of $n$ numbers showing up is noted. What is the probability that among the numbers 1, 2, 3, 4, 5 and 6 only three numbers appear in this list?