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The concept of expected value is fundamental in probability theory. For a negative binomial distribution, it represents the average number of failures expected before achieving a set number of successes. With this specific distribution, you are interested in an experiment repeated until a certain event occurs.To calculate the expected value \(E(X)\) for a negative binomial distribution, you use the formula:\[E(X) = \frac{r(1-p)}{p}\]Here, \(r\) is the number of successes you want, and \(p\) is the probability of success on each trial. In the given problem, \(r = 4\) and \(p = 0.2\). Plugging these values into the formula gives:\[E(X) = \frac{4(1-0.2)}{0.2} = \frac{4 \times 0.8}{0.2} = 16\]This means, on average, you expect 16 failures before achieving 4 successes.

Calculating the probability of a negative binomial variable taking a specific value involves using a formula that reflects the scenario of having a certain number of failures before reaching the desired number of successes.For a negative binomial distribution, the probability \(P(X=k)\) is calculated using:\[P(X=k) = \binom{k+r-1}{r-1} p^r (1-p)^k\]This calculation accounts for different possible failure-success sequences. Let’s explore how this works with the given values:- **For \(X=20\):** \[ P(X=20) = \binom{23}{3} (0.2)^4 (0.8)^{20} \] After simplifying, you get approximately 0.0323. - **For \(X=19\):** \[ P(X=19) = \binom{22}{3} (0.2)^4 (0.8)^{19} \] This results in approximately 0.0355. - **For \(X=21\):** \[ P(X=21) = \binom{24}{3} (0.2)^4 (0.8)^{21} \] Which simplifies to around 0.0294.These probabilities collectively help verify trends or tendencies in a distribution by checking how likely specific outcomes are.

The mode of a distribution represents the most likely or frequent value. For the negative binomial distribution, identifying the mode helps in understanding where the highest probability is concentrated.In a negative binomial distribution, the mode can be determined using:\[k = r + \frac{1-p}{p}\]Then the mode is \((k-1)\). With the provided values, \(r=4\) and \(p=0.2\):\[k = 4 + \frac{0.8}{0.2} = 8\]Thus, the mode is:\[8-1=7\]Therefore, the most likely value for the number of failures before reaching 4 successes is 7. This reflects where the distribution is most concentrated and gives insights into expectations versus reality for this specific probability model.