Sure, let's go through this step by step:
### Step 1: Understand the problem and set up the scenario
Emile has a bag that contains 6 white buttons and 2 blue buttons. He randomly takes 3 buttons out of this bag without replacement. We need to find the probability that all 3 buttons he picks are white.
### Step 2: Determine the total number of buttons in the bag
Total number of buttons = 6 white + 2 blue = 8 buttons.
### Step 3: Calculate the total number of ways to choose 3 buttons from 8
To find the number of ways to choose 3 buttons out of 8, we use the combination formula:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
Where [tex]\( n \)[/tex] is the total number of items, and [tex]\( k \)[/tex] is the number of items to choose.
So, the number of ways to choose 3 buttons from 8 is:
[tex]\[
\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56
\][/tex]
### Step 4: Calculate the number of ways to choose 3 white buttons from 6
Similarly, the number of ways to choose 3 white buttons from 6 is:
[tex]\[
\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
\][/tex]
### Step 5: Determine the probability that all 3 buttons are white
The probability is obtained by dividing the number of favorable outcomes by the total number of possible outcomes.
The favorable outcomes are the ways to choose 3 white buttons out of 6, which we calculated as 20. The total number of possible outcomes is choosing any 3 buttons out of 8, which we calculated as 56.
Thus, the probability is:
[tex]\[
P(\text{3 white buttons}) = \frac{\text{Number of ways to choose 3 white buttons}}{\text{Total number of ways to choose any 3 buttons}} = \frac{20}{56}
\][/tex]
### Step 6: Simplify the fraction
[tex]\[
\frac{20}{56} = \frac{5}{14} \approx 0.35714285714285715
\][/tex]
So, the probability that all 3 buttons Emile picks are white is approximately [tex]\(0.3571\)[/tex] (rounded to four decimal places).
