# Lucky Envelopes

Alignments to Content Standards: S-CP.A.3

There are four red envelopes, four blue envelopes, and four \$1 bills, which will be placed in four of the eight envelopes. Define the event$A$as “you pick a lucky envelope (one that has a \$1 bill in it)” and event $B$ as “you pick a blue envelope”.

1. Suppose one \$1 bill is placed in a blue envelope, and the three remaining \$1 bills are placed in three red envelopes.

1. If you choose one envelope at random, what is the probability that you pick a lucky envelope? How would you write this probability symbolically (using letters $A$ and/or $B$)?
2. If you know that the envelope you picked is blue, what is the probability that you picked a lucky envelope? How would you write this probability symbolically (using letters $A$ and/or $B$)?
3. Did knowing that the envelope is blue change the probability of getting a lucky envelope?

## Solution

1. Out of 8 envelopes, 4 have \$1 bills in them. So the probability of picking a lucky envelope (with a \$1 bill) is $\frac48=\frac12$. Symbolically we write this as $P(A)=\frac12$.
2. In this part, we only consider blue envelopes. Out of 4 blue envelopes, only one has a \$1 bill in it. So the probability of picking the lucky envelope is$\frac14$. This is a conditional probability: the probability that the envelope is lucky given that the envelope is blue. Symbolically we write this as$P(A | B)=\frac14$. 3. Yes, knowing that the envelope picked was blue changed the probability that the envelope is a lucky envelope. It decreased the probability of picking a lucky envelope from$\frac12$to$\frac14$. 1. Out of 8 envelopes, 4 have \$1 bills in them. So the probability of picking a lucky envelope (with a \$1 bill) is$\frac48=\frac12$. Symbolically we write this as$P(A)=\frac12$. 2. In this part, we only consider blue envelopes. Out of 4 blue envelopes, two have \$1 bills in them. So the probability of picking a lucky envelope is $\frac24=\frac12$. This is a conditional probability: the probability that the envelope is lucky given that the envelope is blue. Symbolically we write this as $P(A | B)=\frac12$.
3. No, knowing that the envelope picked was blue did not change the probability that the envelope is a lucky envelope. Either way, the probability of getting a lucky envelope is $\frac12$.

1. In part (a), knowing that the envelope was blue (event $B$) changed the probability that the envelope was a lucky envelope (event $B$) from $\frac12$ to $\frac14$. Therefore, $A$ and $B$ are not independent events.
2. In part (b), knowing that the envelope was blue (event $B$) did not change the probability that the envelope was a lucky envelope (event $B$). Therefore, $A$ and $B$ are independent events.
3. If the events $E$ and $F$ are independent, the definition of independence implies that $P(E)=P(E | F)$.