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  • Assume for simplicity that N people, all born in April (a month of 30 days), are collected in a room. Consider the event of at least two people in the room being born on the same date of

Assume for simplicity that N people, all born in April (a month of 30 days), are collected in a room. Consider the event of at least two people in the room being born on the same date of the month, even if in different year, e.g. 1980 and 1985. What is the smallest N so that the probability of this event exceeds 0.5 ?

Option: 1

20

 


Option: 2

7


Option: 3

15


Option: 4

16


Answers (1)

P(at least two people born on same date) = 1-P(none born on same date)

For                \text{N=2}

Required Probability 

                      \mathrm{=1-\frac{30 C_1}{30 C_1} \times \frac{29 C_1}{30 C_1}=\frac{1}{30} \times \frac{1}{2}}

For                \text{N=3}

Required Probability 

                         \mathrm{=1-\frac{30 C_1}{30 C_1} \times \frac{29 C_1}{30 C_1} \times \frac{28 C_1}{30 C_1} \times \frac{1}{2}}

Similarly for    \text{N = 7}

Required Probability

                             \begin{aligned} = & 1-\frac{30}{30} \times \frac{29}{30} \times \frac{28}{30} \times \frac{27}{30} \times \frac{26}{30} \times \frac{25}{30} \times \frac{24}{30} \\ \end{aligned}

                             \begin{aligned} & =1-0.4692 \\ & =0.5308>\frac{1}{2} \end{aligned}

\Rightarrow   Smallest value of \text{N = 7}

Posted by

Sumit Saini

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