Question

katie listens to a variety of music, 55% of which is country. if she is making a playlist of ten songs, what is the probability that at most three songs are country?

Explanation:

Step1: Identify the distribution

This is a binomial probability problem. Let \( n = 10 \) (number of trials, songs), \( p = 0.55 \) (probability of success, a song being country), and we want \( P(X\leq3) \) where \( X \) is the number of country songs. The binomial probability formula is \( P(X = k)=\binom{n}{k}p^{k}(1 - p)^{n - k} \), and \( P(X\leq3)=P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3) \).

Step2: Calculate \( P(X = 0) \)

\( \binom{10}{0}=1 \), \( p^{0}=1 \), \( (1 - 0.55)^{10 - 0}=0.45^{10}\approx0.0003405 \). So \( P(X = 0)=1\times1\times0.0003405 = 0.0003405 \).

Step3: Calculate \( P(X = 1) \)

\( \binom{10}{1}=\frac{10!}{1!(10 - 1)!}=10 \), \( p^{1}=0.55 \), \( (1 - 0.55)^{9}=0.45^{9}\approx0.0007567 \). So \( P(X = 1)=10\times0.55\times0.0007567\approx0.004162 \).

Step4: Calculate \( P(X = 2) \)

\( \binom{10}{2}=\frac{10!}{2!(10 - 2)!}=45 \), \( p^{2}=0.55^{2}=0.3025 \), \( (1 - 0.55)^{8}=0.45^{8}\approx0.0016816 \). So \( P(X = 2)=45\times0.3025\times0.0016816\approx0.02327 \).

Step5: Calculate \( P(X = 3) \)

\( \binom{10}{3}=\frac{10!}{3!(10 - 3)!}=120 \), \( p^{3}=0.55^{3}=0.166375 \), \( (1 - 0.55)^{7}=0.45^{7}\approx0.003737 \). So \( P(X = 3)=120\times0.166375\times0.003737\approx0.07464 \).

Step6: Sum the probabilities

\( P(X\leq3)=0.0003405 + 0.004162+0.02327 + 0.07464\approx0.1024 \).

Answer:

The probability is approximately \( 0.1024 \) (or 10.24%).