a test coonsist of 10 true/false questions. To pass the test a student must answer atleast 6 questions correctly. If a student guesses on each question, what is the probability that the student will pass the test...
show how you got the answer please. thank you..
+118724 They have to get 6, 7,8,9,or 10 correct
$$\\P=10C6(0.5)^6(0.5^4)+10C7(0.5)^7(0.5)^3+10C8(0.5)^8(0.5)^2+10C9(0.5)^9(0.5)^1+10C10(0.5)^{10}(0.5)^0\\\\
P=10C6(0.5)^{10}+10C7(0.5)^{10}+10C8(0.5)^{10}+10C9(0.5)^{10}+10C10(0.5)^{10}\\\\
P=10C6(0.5)^{10}+10C7(0.5)^{10}+10C8(0.5)^{10}+10*(0.5)^{10}+(0.5)^{10}\\\\$$
$${\left({\frac{{\mathtt{10}}{!}}{{\mathtt{6}}{!}{\mathtt{\,\times\,}}({\mathtt{10}}{\mathtt{\,-\,}}{\mathtt{6}}){!}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{0.5}}}^{{\mathtt{10}}}{\mathtt{\,\small\textbf+\,}}{\left({\frac{{\mathtt{10}}{!}}{{\mathtt{7}}{!}{\mathtt{\,\times\,}}({\mathtt{10}}{\mathtt{\,-\,}}{\mathtt{7}}){!}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{0.5}}}^{{\mathtt{10}}}{\mathtt{\,\small\textbf+\,}}{\left({\frac{{\mathtt{10}}{!}}{{\mathtt{8}}{!}{\mathtt{\,\times\,}}({\mathtt{10}}{\mathtt{\,-\,}}{\mathtt{8}}){!}}}\right)}{\mathtt{\,\times\,}}{\left({\mathtt{0.5}}\right)}^{{\mathtt{10}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{10}}{\mathtt{\,\times\,}}{{\mathtt{0.5}}}^{{\mathtt{10}}}{\mathtt{\,\small\textbf+\,}}{{\mathtt{0.5}}}^{{\mathtt{10}}} = {\frac{{\mathtt{193}}}{{\mathtt{512}}}} = {\mathtt{0.376\: \!953\: \!125}}$$
so the probability of passing is 0.376953125
+118724 They have to get 6, 7,8,9,or 10 correct
$$\\P=10C6(0.5)^6(0.5^4)+10C7(0.5)^7(0.5)^3+10C8(0.5)^8(0.5)^2+10C9(0.5)^9(0.5)^1+10C10(0.5)^{10}(0.5)^0\\\\
P=10C6(0.5)^{10}+10C7(0.5)^{10}+10C8(0.5)^{10}+10C9(0.5)^{10}+10C10(0.5)^{10}\\\\
P=10C6(0.5)^{10}+10C7(0.5)^{10}+10C8(0.5)^{10}+10*(0.5)^{10}+(0.5)^{10}\\\\$$
$${\left({\frac{{\mathtt{10}}{!}}{{\mathtt{6}}{!}{\mathtt{\,\times\,}}({\mathtt{10}}{\mathtt{\,-\,}}{\mathtt{6}}){!}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{0.5}}}^{{\mathtt{10}}}{\mathtt{\,\small\textbf+\,}}{\left({\frac{{\mathtt{10}}{!}}{{\mathtt{7}}{!}{\mathtt{\,\times\,}}({\mathtt{10}}{\mathtt{\,-\,}}{\mathtt{7}}){!}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{0.5}}}^{{\mathtt{10}}}{\mathtt{\,\small\textbf+\,}}{\left({\frac{{\mathtt{10}}{!}}{{\mathtt{8}}{!}{\mathtt{\,\times\,}}({\mathtt{10}}{\mathtt{\,-\,}}{\mathtt{8}}){!}}}\right)}{\mathtt{\,\times\,}}{\left({\mathtt{0.5}}\right)}^{{\mathtt{10}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{10}}{\mathtt{\,\times\,}}{{\mathtt{0.5}}}^{{\mathtt{10}}}{\mathtt{\,\small\textbf+\,}}{{\mathtt{0.5}}}^{{\mathtt{10}}} = {\frac{{\mathtt{193}}}{{\mathtt{512}}}} = {\mathtt{0.376\: \!953\: \!125}}$$
so the probability of passing is 0.376953125
