To determine the probability of getting exactly 1 tail when a fair coin is tossed 3 times, we can use the binomial probability formula. The binomial probability formula is given by:
[tex]\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} \][/tex]
Where:
- [tex]\( n \)[/tex] is the number of trials (coin tosses),
- [tex]\( k \)[/tex] is the number of desired successful outcomes (tails),
- [tex]\( p \)[/tex] is the probability of success on a single trial (getting tails in one toss),
- [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, calculated as [tex]\(\frac{n!}{k!(n-k)!}\)[/tex].
Let's break it down step-by-step:
1. Number of Trials (n): The coin is tossed 3 times, so [tex]\( n = 3 \)[/tex].
2. Desired Successes (k): We want exactly 1 tail, so [tex]\( k = 1 \)[/tex].
3. Probability of Success (p): It's a fair coin, so the probability of getting tails in a single toss is [tex]\( p = 0.5 \)[/tex].
4. Binomial Coefficient ([tex]\(\binom{n}{k}\)[/tex]): We need to calculate [tex]\(\binom{3}{1}\)[/tex]:
[tex]\[ \binom{3}{1} = \frac{3!}{1!(3-1)!} = \frac{3!}{1! \cdot 2!} = \frac{3 \cdot 2 \cdot 1}{1 \cdot 2 \cdot 1} = 3 \][/tex]
5. Putting it all together: Substituting [tex]\( n = 3 \)[/tex], [tex]\( k = 1 \)[/tex], and [tex]\( p = 0.5 \)[/tex] into the binomial probability formula:
[tex]\[ P(X = 1) = \binom{3}{1} \cdot (0.5)^1 \cdot (1-0.5)^{3-1} \][/tex]
[tex]\[ P(X = 1) = 3 \cdot (0.5) \cdot (0.5)^2 \][/tex]
[tex]\[ P(X = 1) = 3 \cdot 0.5 \cdot 0.25 \][/tex]
[tex]\[ P(X = 1) = 3 \cdot 0.125 \][/tex]
[tex]\[ P(X = 1) = 0.375 \][/tex]
Thus, the probability of getting exactly 1 tail when a fair coin is tossed 3 times is [tex]\( 0.375 \)[/tex].
To the nearest thousandth, the probability of getting exactly 1 tail in 3 coin tosses is:
[tex]\[ \boxed{0.375} \][/tex]
