Question

Each of the two Shooters fires 10 shots at their own target. The chance that they will score the same number of hits if the probability of hitting in each shot is 0.5, is given by

• A. $\displaystyle\prod_{ r =1}^{10}\left(\frac{2 r -1}{2 r }\right)$
• B. $\displaystyle\prod_{r=1}^{10}\left(\frac{r+1}{4 r}\right)$
• C. $\frac{{ }^{20} C _{10}}{4^{10}}$
• D. $\frac{(10) !}{\displaystyle\prod_{r=1}^{10}(2 r-1)}$

Answer 1

Answer: (A), (C)

$P \text { (they score the same number of hits) }=[\text { (He score none) } \text { and (other also score none) on so on } $
$\text { (He score one) and (He also score one) }+\ldots . . . \ldots \ldots \ldots . . .]$
$=\left({ }^{10} C _0 \frac{1}{2^{10}}\right)\left({ }^{10} C _0 \frac{1}{2^{10}}\right)+\left({ }^{10} C _1 \frac{1}{2^{10}}\right)\left({ }^{10} C _1 \frac{1}{2^{10}}\right)+\ldots \ldots $
$=\frac{ C _0^2+ C _1^2+ C _2^2+\ldots \ldots .+ C _{10}^2}{2^{20}}=\frac{{ }^{20} C _{10}}{4^{10}}=\frac{(20) !}{10 ! \cdot 10 ! \cdot 4^{10}} \Rightarrow( C ) $