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  • A coin is tossed n times. The probability of getting at least one head is greater than that of getting at least two tails by <img alt="\frac{5}{32}" src="https://entrancecorner.oncodecogs.com/

A coin is tossed n times. The probability of getting at least one head is greater than that of getting at least two tails by \frac{5}{32}. Then n is

Option: 1

5


Option: 2

10


Option: 3

15


Option: 4

None of these


Answers (1)

best_answer

The probability of getting at least one head

                 =1- probability of getting no heads \mathrm{=1-{ }^n C_0\left(\frac{1}{2}\right)^0 \cdot\left(\frac{1}{2}\right)^n=1-\frac{1}{2^n}}

The probability of getting at least two tails
                 =1- probability of getting no tails - probability of getting 2 tails

                  \mathrm{=1-{ }^n C_n\left(\frac{1}{2}\right)^n \cdot\left(\frac{1}{2}\right)^0-{ }^n C_{n-1}\left(\frac{1}{2}\right)^{n-1} \cdot \frac{1}{2}=1-\frac{1}{2^n}-n \frac{1}{2^n}}

From the question, \mathrm{\left(1-\frac{1}{2^n}\right)-\left(1-\frac{1+n}{2^n}\right)=\frac{5}{32}}

       Or \mathrm{\frac{n+1}{2^n}-\frac{1}{2^n}=\frac{5}{32} \quad \text { or } \frac{n}{2^n}=\frac{5}{32} \quad \Rightarrow n=5 \text {. }}

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Anam Khan

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