I am confused about the following two scenarios:
Out of a bag of 3 apples and 3 oranges, you pick 2 items.
1) What is the probability that you will have 2 apples?
2) What is the probability that you will have 1 apple and 1 orange?
My attempt:
1)
$$
\begin{aligned}
P(\mbox{2 apples})
&= P(\mbox{1st apple}) \times P(\mbox{2nd apple}) \\
&= \frac{3}{6} \times \frac{2}{5}.
\end{aligned}
$$
2)
$$
\begin{aligned}
P(\mbox{1 apple and 1 orange})
&= P(\mbox{1st apple}) \times P(\mbox{2nd orange}) \\
& + P(\mbox{1st orange}) \times P(\mbox{2nd apple}) \\
&= \frac{3}{6} \times \frac{2}{5} \times 2.
\end{aligned}
$$
My confusion is with case number 1: why you don't need to multiply the result by 2? Since your first pick could be apple #1, #2, #3.
Best Answer
Consider all of the $6\times 5$ ways to pick two pieces of fruit. That's $30$: $$\boxed{\begin{array}{|l|ccc:ccc|}\hline ~ & A_1 & A_2 & A_3 & O_1 & O_2 & O_3 \\ \hline A_1 & \times & \color{green}{A_1,A_2} & \color{green}{A_1,A_3} & \color{blue}{A_1,O_1} & \color{blue}{A_1,O_2} & \color{blue}{A_1,O_3} \\ A_2 & \color{green}{A_2,A_1} & \times & \color{green}{A_2,A_3} & \color{blue}{A_2,O_1} & \color{blue}{A_2,O_2} & \color{blue}{A_2,O_3} \\ A_3 & \color{green}{A_3,A_1} & \color{green}{A_3,A_2} & \times & \color{blue}{A_3,O_1} & \color{blue}{A_3,O_2} & \color{blue}{A_3,O_3} \\ \hdashline O_1 & \color{indigo}{O_1,A_1} & \color{indigo}{O_1,A_2} & \color{indigo}{O_1,A_3} & \times & \color{red}{O_1,O_2} & \color{red}{O_1,O_3} \\ O_2 & \color{indigo}{O_2,A_1} & \color{indigo}{O_2,A_2} & \color{indigo}{O_2,A_3} & \color{red}{O_2,O_1} & \times & \color{red}{O_2,O_3} \\ O_3 & \color{indigo}{O_3,A_1} & \color{indigo}{O_3,A_2} & \color{indigo}{O_3,A_3} & \color{red}{O_3,O_1} & \color{red}{O_3,O_2} & \times \\ \hline \end{array}}$$
The ways to pick two apples are in the green quarter (upper left). There are $3\times 2$ of them; that is $6$ of $30$
The ways to pick an apple and an orange are in the blue quarter, but also in the indigo quarter (upper-right and lower-left). There are $3\times 3+3\times 3$ of them; that's $18$ of $30$
The final quarter are ways to pick two oranges. Again, just $3\times 2$ of these; that's $6$ of $30$.