91影视

A school band contains:

10 students from the freshman class.

20 students from sophomore

30 students from the junior class.

40 students from the senior class.

15 students are selected from the class randomly.

The probability that at least one student will be selected from each of the four classes is expressed as,

\(\begin{array}{c}{\bf{P}}\left( {\bf{E}} \right){\bf{ = 1 - P}}\left( {{\bf{The}}\;{\bf{band}}\;{\bf{does}}\;{\bf{not}}\;{\bf{contain}}\;{\bf{any}}\;{\bf{one}}\;{\bf{class}}\;{\bf{member}}} \right)\\{\bf{ = 1 - P}}\left( {{{\bf{E}}^{\bf{c}}}} \right)\end{array}\)

The favourable outcomes that a freshman is not selected is,\(^{100 - 10}{C_{15}}{ = ^{90}}{C_{15}}\).

The favourable outcomes that a sophomore is not selected is,\(^{80}{C_{15}}\).

The favourable outcomes that a junior is not selected is,\(^{70}{C_{15}}\).

The favourable outcomes that a senior is not selected is,\(^{60}{C_{15}}\).

Total number of 15 band members from 100 students is,\(^{100}{C_{15}}\).

The probability that student is not selected from exactly one class,

\(\begin{aligned}{}P\left( {{E^c}} \right) &= \frac{{^{90}{C_{15}}{ + ^{80}}{C_{15}}{ + ^{70}}{C_{15}}{ + ^{ + 60}}{C_{15}}}}{{^{100}{C_{15}}}}\\ &= 0.2100\end{aligned}\)\(\)

Thus, the probability that at least one student will be selected from each of the four classes is,

\(\begin{aligned}{}P\left( E \right) &= 1 - 0.2100\\ &= 0.7900\end{aligned}\)

Thus, the required probability is 0.7900.