Let n = 25, p = 0.4, q = 1-0.4=0.6.
The formula for the binomial probability is as follows:
[tex]P(x)=_nC_xp^x(1-p)^{n-x}[/tex]
To obtain the probability of at least 10, solve for P(10)+P(11)+P(12)+...+P(25).
Thus, we have the following.
[tex]\begin{gathered} P(x\ge10)=_{25}C_{10}(0.4)^{10}(0.6)^{25-10}+_{25}C_{11}(0.4)^{11}(0.6)^{25-11}+\cdots+_{25}C_{25}(0.4)^{25}(0.6)^{25-25} \\ =(3268760)(0.4)^{10}(0.6)^{15}+(4457400)(0.4)^{11}(0.6)^{14}+\cdots+(1)(0.4)^{25}(0.6)^0 \\ \approx0.57538298232 \\ \approx0.5754 \end{gathered}[/tex]
Therefore, the probability that at least 10 of them prefer the news online is approximately 0.5754 or 57.54%.
