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Chapter 1: Q11E (page 51)

Suppose that 10 cards, of which five are red and five are green, are placed at random in 10 envelopes, of which five are red and five are green. Determine the probability that exactly x envelopes will contain a card with a matching colour (x = 0, 1, ... , 10).

Short Answer

Expert verified

The probability that exactly x envelopes will contain a card with a matching colour\(\frac{{\left( {^5{C_{\frac{r}{2}}}} \right)\left( {^5{C_{_{\frac{r}{2}}}}} \right)}}{{10!}}\;\;;\;\forall r = 0,2,...,10\). In terms of \(\frac{{\left( {^5{C_x}} \right)\left( {^5{C_x}} \right)}}{{10!}}\;;\;\forall x\left( { = 2r} \right) = 0,1,...,10\)

Step by step solution

01

Given information

Number of red and green coloured envelopes is 5 each. Also, the number of red and green coloured letters is 5 each.

02

Compute the counts

The total number of favourable outcomes are \(10!\)

On selecting 5 red or 5 green envelopes, number of red or green envelopes that could be placed in right colour combination is,

\(\begin{array}{l}{\bf{r = 0,1,2,3,4,5}}\\{\bf{g = 0,1,2,3,4,5}}\end{array}\)

Hence,

\(\begin{array}{c}x = 2r\\ = 2g\end{array}\)

And\(x = 0,2,4,...,10\).

The probability that x envelope would contain the right colour combination card is,

\(\frac{{^5{C_x}^{5 - x}{C_x}}}{{10!}} = \frac{{^5{C_x}^5{C_x}}}{{10!}}\)

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Most popular questions from this chapter

Let S be a given sample space and let \({A_1},{A_2},.....\) bean infinite sequence of events. For \(n = 1,2,..........\), let \({B_n} = \bigcup\limits_{i = n}^\infty {{A_i}} \)and let \({C_n} = \bigcap\limits_{i = n}^\infty {{A_i}} \)

a. Show that \({B_1} \supset {B_2} \supset \ldots \ldots \) and that \({C_1} \subset {C_2} \subset \ldots \ldots \).

b. Show that an outcome in S belongs to the event\(\bigcap\limits_{n = 1}^\infty {{B_n}} \) if and only if it belongs toan infinite number of the events \({A_1},{A_2},.....\).

c. Show that an outcome in S belongs to the event \(\bigcup\limits_{n = 1}^\infty {{C_n}} \)if and only if it belongs to all the events\({A_1},{A_2},.....\) except possibly a finite number of those events.

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