Question

A “friend” offers you the following “deal.” For a $\backslash (10$fee, you may pick an envelope from a box containing $100$seemingly identical envelopes. However, each envelope contains a coupon for a free gift.

• Ten of the coupons are for a free gift worth$\backslash )6$.

• Eighty of the coupons are for a free gift worth $\backslash (8$.

• Six of the coupons are for a free gift worth $\backslash )12$.

• Four of the coupons are for a free gift worth $\$40.$

Based upon the financial gain or loss over the long run, should you play the game?

a. Yes, I expect to come out ahead in money.

b. No, I expect to come out behind in money.

c. It doesn’t matter. I expect to break even.

Short answer

(b) No, I should not play, I expect to come out behind in money.

Step-by-step solution

Given information 

There are$100$identical envelopes and have to pick one envelope cost of $\$10$.

Each envelope contains a gift. They are:

Ten envelopes contain $\$6$gift.

Eighty envelopes contain $\$8$gift.

Six envelopes contain $\$12$gift.

Four envelopes contain$\$40$gift.

Explanation

X represents the net profit.

$X$	$P\left(X\right)$	$X.P\left(x\right)$
$-4$	$10/100=0.10$	$-0.4$
$-2$	$80/100=0.80$	$-1.6$
$2$	$6/100=0.06$	$0.12$
$30$	$4/100=0.04$	$1.2$
Total	$1$	$-0.68$

As we can see, if we play this game, we will lose money, so I expect to lose money.