Pick a box, any box

Probability Level 4

A room contains 150 boxes. The boxes are identical except for colour. Each box contains a card which has one of three messages printed on it.

In 78 of the boxes the card says “No Prize”. In 66 of the boxes the card says “Winner $20”. And in 6 of the boxes the card says “Winner $100”. Contestants have been assigned a number corresponding to when they will make their selection.

On a turn, each contestant gets to randomly choose 2 boxes. Once a box is chosen it is removed from the room.

You are the second contestant.

If the probability that you will select at least one of the boxes containing a $100 prize can be expressed as $\dfrac{a}{b}$ , where $a$ and $b$ are coprime positive integers , find $a+b$ .

The answer is 4018.

2 solutions

Geoff Pilling
Aug 15, 2016

Since it isn't revealed what the first contestant chose, then, by symmetry, you both have the same probability of having the $100 prize, namely,

$P = 1 - (\frac{144}{150})(\frac{143}{149}) = \frac{293}{3725}$ $293 + 3725 = \boxed{4018}$

Didnt expect this to be that simple :0

Armain Labeeb - 4 years, 9 months ago

Armain Labeeb
Aug 2, 2016

There are $^{ 150 }{ C }_{ 4 }=486246600$ ways to pick the first $4$ boxes.

Six of the boxes contain a $\$100$ card and $150 - 6 = 144$ do not contain a $\$100$ card. Once a $\$100$ card is selected the number of available $\$100$ cards decreases by $1$ . Once any card other than a $\$100$ card is selected the number of such cards decreases by $1$ .

I will present a simplified solution in a chart to determine the total number of ways for contestant $2$ to select at least one $\$100$ card. If for any selection a $\$100$ card is selected, we will represent the selection with a G . If for any selection a $\$100$ card is not selected, I will represent the selection with an $N$ . So GGNG represents the possibility that the first contestant picked two $\$100$ cards and the second contestant picked a non- $\$100$ card then a $\$100$ card. The possible cases are shown in the chart.

$\text{Contestant 1 Selection}$	$\text{Contestant 2 Selection}$	$Calculations$	$\text{No. of Possiblities}$
$GG$	$GN$	$6\times5\times4\times144$	$17280$
$GG$	$NG$	$6\times5\times144\times4$	$17280$
$GG$	$GG$	$6\times5\times4\times3$	$360$
$GN$	$GN$	$6\times144\times5\times143$	$617760$
$GN$	$NG$	$6\times144\times143\times5$	$617760$
$GN$	$GG$	$6\times144\times5\times4$	$17280$
$NG$	$GN$	$144\times6\times5\times143$	$617760$
$NG$	$NG$	$144\times6\times143\times5$	$617760$
$NG$	$GG$	$144\times6\times5\times4$	$17280$
$NN$	$GN$	$144\times143\times6\times142$	$17544384$
$NN$	$NG$	$144\times143\times142\times6$	$17544384$
$NN$	$GG$	$144\times143\times6\times5$	$617760$
		$\text{Total Possibilities}$	$38247048$

$\large\begin{aligned} & \therefore \quad & \frac { a }{ b } & =\frac { 38247048 }{ 486246600 } & =&\frac { 293 }{ 3725 } \\ & \Longrightarrow & a+b & =293+3725 & =&\boxed{4018} \end{aligned}$

Pick a box, any box

The answer is 4018.

2 solutions

0 pending reports