Negative Binomial Distribution - Flashcard set - Probability

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It rains in an area with a probability of 20% each day. What is the probability that it rains for the third time on the eighth day?

The probability, using the Negative Binomial Distribution, is $P(X=k) = \binom{k-1}{r-1}p^r(1-p)^{k-r}$ , with $k=8$ , $r=3$ , and $p=0.2$ . So, $P(X=8) = \binom{7}{2}(0.2)^3(0.8)^5$ .

A store has a 40% chance of selling a particular item each day. What's the probability the 5th item is sold on the 9th day?

Use the formula: $P(X=k) = \binom{k-1}{r-1}p^r(1-p)^{k-r}$ , where $k=9$ , $r=5$ , and $p=0.4$ , leading to $P(X=9) = \binom{8}{4}(0.4)^5(0.6)^4$ .

A basketball player continues shooting until they've made 10 baskets. If their success rate is 60%, what's the probability they will make their 10th basket on the 15th shot?

Compute the probability with $P(X=k) = \binom{k-1}{r-1}p^r(1-p)^{k-r}$ where $k=15$ , $r=10$ , and $p=0.6$ . $P(X=15) = \binom{14}{9}(0.6)^{10}(0.4)^5$ .

A biologist observes a particular species of frog that has a 10% chance of calling within a minute. What's the probability the frog's 4th call is heard after 20 minutes?

Use the Negative Binomial Distribution: $P(X=k) = \binom{k-1}{r-1}p^r(1-p)^{k-r}$ , for $k=20$ , $r=4$ , and $p=0.1$ . So, $P(X=20) = \binom{19}{3}(0.1)^4(0.9)^{16}$ .

In a series of job interviews, an applicant has a 30% chance of receiving a job offer after each interview. How likely is it they receive their third offer following the seventh interview?

Calculate with: $P(X=k) = \binom{k-1}{r-1}p^r(1-p)^{k-r}$ , where $k=7$ , $r=3$ , and $p=0.3$ . Therefore, $P(X=7) = \binom{6}{2}(0.3)^3(0.7)^4$ .

You flip a coin until you get 3 heads. What's the probability the third head will be on the sixth flip?

The probability is calculated using the Negative Binomial Distribution formula: $P(X=k) = \binom{k-1}{r-1}p^r(1-p)^{k-r}$ where $k=6$ , $r=3$ , and $p=0.5$ for a fair coin. $P(X=6) = \binom{5}{2}(0.5)^3(0.5)^3$ .

If there's a 5% chance to catch a rare fish each time you cast a line, what's the probability you catch your second rare fish on your twentieth cast?

Calculate with: $P(X=k) = \binom{k-1}{r-1}p^r(1-p)^{k-r}$ , where $k=20$ , $r=2$ , and $p=0.05$ . Thus, $P(X=20) = \binom{19}{1}(0.05)^2(0.95)^{18}$ .

A logistic company delivers packages with a 90% success rate. What's the likelihood that the 100th successful delivery occurs on the 150th attempt?

Apply the Negative Binomial Distribution: $P(X=k) = \binom{k-1}{r-1}p^r(1-p)^{k-r}$ , for $k=150$ , $r=100$ , and $p=0.9$ . Therefore, $P(X=150) = \binom{149}{99}(0.9)^{100} (0.1)^{50}$ .

A code generates 'pass' with a 25% chance on each run. Calculate the likelihood that the 4th 'pass' comes on the 10th run.

Apply the Negative Binomial Distribution: $P(X=k) = \binom{k-1}{r-1}p^r(1-p)^{k-r}$ , for $k=10$ , $r=4$ , and $p=0.25$ . Hence, $P(X=10) = \binom{9}{3}(0.25)^4(0.75)^6$ .

A scientist needs exactly 7 successful experiments for a valid test sequence. If the success rate is 70% per experiment, what's the probability the 7th success occurs at experiment 12?

Use the Negative Binomial Distribution: $P(X=k) = \binom{k-1}{r-1}p^r(1-p)^{k-r}$ with $k=12$ , $r=7$ , and $p=0.7$ . Thus, $P(X=12) = \binom{11}{6}(0.7)^7(0.3)^5$ .

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