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Mathematics

Probability

Binomial Distribution

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A soccer player makes a goal 80% of the time when taking a penalty kick. What is the probability the player makes exactly 3 out of 5 penalty kicks?

$P(X = 3) = \binom{5}{3} \left(0.80\right)^3 \left(0.20\right)^{5-3}$

An online survey shows that 10% of customers are satisfied with a service. If 200 customers are surveyed, what is the probability that exactly 30 are satisfied?

$P(X = 30) = \binom{200}{30} \left(0.10\right)^{30} \left(0.90\right)^{200-30}$

A brand of chocolate bars includes a golden ticket 1% of the time. How likely is it to find exactly 2 golden tickets in a case of 100 bars?

$P(X = 2) = \binom{100}{2} \left(0.01\right)^2 \left(0.99\right)^{100-2}$

A multiple choice exam has 25 questions. Each has four possible answers with only one correct answer. If a student guesses on each question, what is the probability of guessing exactly 10 correct?

$P(X = 10) = \binom{25}{10} \left(\frac{1}{4}\right)^{10} \left(\frac{3}{4}\right)^{25-10}$

A product line has a 2% defect rate. If 60 products are randomly selected, what is the probability that exactly 1 is defective?

$P(X = 1) = \binom{60}{1} \left(0.02\right)^1 \left(0.98\right)^{60-1}$

Flip a coin 10 times. What is the probability of getting exactly 6 heads?

$P(X = 6) = \binom{10}{6} \left(\frac{1}{2}\right)^6 \left(\frac{1}{2}\right)^{10-6}$

A restaurant's dish is liked by 85% of its patrons. If 20 patrons are randomly selected, what is the probability that 17 or more patrons will like the dish?

$P(X \geq 17) = \sum_{x=17}^{20} \binom{20}{x} \left(0.85\right)^x \left(0.15\right)^{20-x}$

A species of fish has a probability of 70% to survive past its first year. In a pond with 50 such fish, what is the probability that 35 survive the first year?

$P(X = 35) = \binom{50}{35} \left(0.70\right)^{35} \left(0.30\right)^{50-35}$

In a batch of 20 light bulbs, 5 are defective. What is the probability that 2 randomly chosen bulbs are defective?

$P(X = 2) = \binom{20}{2} \left(\frac{5}{20}\right)^2 \left(\frac{15}{20}\right)^{20-2}$

For a certain disease, a test has a 95% chance of returning a true positive. If 100 people known to have the disease are tested, what's the probability that exactly 90 test positive?

$P(X = 90) = \binom{100}{90} \left(0.95\right)^{90} \left(0.05\right)^{100-90}$

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