A man takes a forward step with probability 0.8 and backward step with probability 0.2. What is the probability that at
the end of 9 steps he is exactly three steps away from the starting point
Answers
Given : A man takes a forward step with probability 0.8 and backward step with probability 0.2.
To Find : the probability that at the end of 9 steps he is exactly three steps away from the starting point
Solution:
3 Steps away can be in 2 cases
6 step forward + 3 backward = 3 step forward
3 step forward + 6 backward = 3 step backward
P ( F) = 0.8 = p
P(B) = 0.2 = q
P(x) = ⁿCₓpˣqⁿ⁻ˣ
n = 9
x = 6 or 3
The probability that at the end of 9 steps he is exactly three steps away from the starting point
= P(6) + P(3)
= ⁹C₆(0.8)⁶(0.2)³ + ⁹C₃(0.8)³(0.2)⁶
0.8 = 4/5 0.2 = 1/5 , ⁹C₆ = ⁹C₃ = 84
= 84 * 4⁶ /5⁹ + 84 * 4³ / 5⁹
= 84 * 4³ ( 4³ + 1) / 5⁹
= 84 * 64 * 65 / 5⁹
= 84 * 64 * 13/5⁸
= 69888/5⁸
Correct option is d) 69888/5⁸
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