Page 339 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes Example 14: Find the probability of throwing 6 at least once in six throws with a single die.

1
Solution: p = probability of throwing 6 with a single die =
6
1 5
q = 1 − =
6 6

1 5
n = 6, p = , q =
6 6
p (at least one six)= 1 – P [none six in 6 throws]

⎡ ⎛⎞ 0 ⎛⎞ 6 ⎞ ⎛⎞ 6
1
5
5
= − 1 6 0 ⎜⎟ ⋅ ⎢ C ⎜⎟ ⎟ = − ⎜⎟
1
⎟
6
6
6
⎢ ⎣ ⎝⎠ ⎝⎠ ⎠ ⎝⎠
Example 15: Three dice are thrown. What is the probability that at least one of the numbers turning
up being greater than 4 ?
Solution: p = probability of a number greater than 4 (i.e., 5 and 6) in a throw of one die
1 1 2 1
= + = =
6 6 6 3
q = 1 − 1 = 2
3 3
1 2
∴ n = 3, p = , q =
3 3
P (at least one number greater than 4) = 1 – P (none of the number greater than 4)

⎡ ⎛⎞ 0 ⎛ 2 ⎞ 3 ⎤
1
= 1 − 3 C 0 ⎜⎟ ⋅ ⎢ ⎜ ⎟ ⎥
3
⎢ ⎝⎠ ⎝ 3 ⎠ ⎥ ⎣ ⎦
⎛⎞ 3 2 2 2 19
2
1
= − ⎜⎟ = 1 − × × =
⎝⎠ 3 3 3 27
3
Example 16: A and B play for a prize of Rs. 1000. A is to throw a die first and is to win if he throws
6. If he fails B is to throw and is to win if throws 6 or 5. If he fails A is to throw again
and to win if he throws 6, 5 or 4 and so on. Find their respective expectations.
1
Solution: Probability of A’s winning in the 1st throw (i.e., he throws 6) =
6

5 2 5
Probability of B’s winning in the 2nd throw (i.e., he throws 6 or 5) = × =
6 6 18
5 4 3 5
Probability of A’s winning in the 3rd throw (6 or 5 or 4) = ×× =
6 6 6 18

5 4 3 4 5
Probability of B’s winning in the 4th throw (6 or 5 or 4 or 3) = × × × =
6 6 6 6 27
Probability of A’s winning in the 5th throw (6 or 5 or 4 or 3 or 2)

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