1. (a). A bag contains 6 red, 5 white, 4 black ball. If two balls are drawn then (1) probability that none of them is red (2). probability that one is white and one is black. (b). An experimenter conduct a coin experiment in a way that he ll stop the experiment when he ll get first success and success of experiment counts when tail appears. Which distribution is appropriate in this experiment and give the explanation. Also determine the expression \( P (X \leq 7) \) for the given experiment. (6)

Question

1. (a). A bag contains 6 red, 5 white, 4 black ball. If two balls are drawn then (1) probability that none of them is red (2). probability that one is white and one is black. 

2. An experimenter conduct a coin experiment in a way that he ll stop the experiment when he ll get first success and success of experiment counts when tail appears. Which distribution is appropriate in this experiment and give the explanation. Also determine the expression ( P (X <= 7) for the given experiment. 

Answer 1

(1) n(R) = 6, n(W) = 5, n(B) = 4

n(S) = 6 + 5 + 4 = 15

∵ Two balls are drawn.

(a) Probability that none of them is red (it means both balls are non -red)

\(= \frac{9C_2}{15C_2}\)

\( = \frac{9\times 8}{15\times 14} \)

\( = \frac{12}{35}\)

(b) Probability that one is white and one is black

\(= \frac{5C_1 \times 4C_1}{15C_2}\)

\( = \frac{5\times 4}{\frac{15\times 14}{2}}\)

\(= \frac4{21}\)

(2) P(x = 1) = Probability of tail in one trial = \(\frac12\)

P(x = 2) = Probability of head in first trial and trail in second trial = \(\frac12 \times \frac 12 = \frac14\)

P(x = 3) = Probabilty of head in first two trails and tail in third trial = \(\left(\frac12\right)^2 \times \frac12 = \frac18\)

P(x = 7) = Probability of head in first six trials and tail in seventh trial = \(\left(\frac12\right)^6 \times \left(\frac12\right)= \left(\frac12\right)^7\) 

\(\therefore P(X\le 7) = P(x = 1) + P(x = 2) +... + P(x = 7)\)

\(= \frac12 + \left(\frac12\right)^2 + ....+ \left(\frac12\right)^7\)

\(= \frac{\frac12 \left(1-\left(\frac12\right)^7\right)}{1 - \frac12}\)

\(= 1 - \left(\frac12\right)^7\)

\(= 1- \frac1{128}\)

\(= \frac{127}{128}\)

Comments

Hello sir I&#39;ve 4 more questions can you answer them too please

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2. (a) Let mean is 2 then use the poisson distribution for find the P(X ≤ 4), P(X ≥ 5) and P(X=3). and also find moment generating function M_X=3(t = 3) for same given problem.

(b) X: -2, -1, 0, 1, 2, 3                                            p(x): 0.1, k, 0.2, 2k, 0.3, k (1) Find the value of k, mean and variance. (2). Construct CDF and drar the graph.



3.(a)X: -0, -1, 2, 3
p(x): 0.1, 0.3, 0.5, 0.1
Find probability function of Y=X^2+2X . Also find P(X ≤ 3), P(1