Nana adalah consumer loan officer di Bank Mandiri. Berdasarkan pengalamannya, dia memperkirakan peluang seorang nasabah akan mampu membayar hutangnya adalah $0.30$ . Bulan lalu Nana berhasil mendapatkan $20$ nasabah baru. Berapakah peluang paling tidak tiga nasabah bisa membayar hutangnya ?

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Q. Nana adalah consumer loan officer di Bank Mandiri. Berdasarkan pengalamannya, dia memperkirakan peluang seorang nasabah akan mampu membayar hutangnya adalah

0.30

. Bulan lalu Nana berhasil mendapatkan

20

nasabah baru. Berapakah peluang paling tidak tiga nasabah bisa membayar hutangnya ?

Identify Problem Type: Step $1$ : Understanding the Problem $\newline$ Reasoning: Identify the probability problem type and the relevant parameters. $\newline$ Calculation: Nana estimates the probability of a customer being able to repay their debt as $0.30$ . She has $20$ new customers.
Define Random Variable: Step $2$ : Define the Random Variable $\newline$ Reasoning: Define the random variable $X$ as the number of customers who can repay their debt. $\newline$ Calculation: $X$ follows a binomial distribution, where $n = 20$ (number of trials/customers) and $p = 0.30$ (probability of success on each trial).
Calculate Probability: Step $3$ : Calculate the Probability $\newline$ Reasoning: Calculate the probability that at least three customers can repay their debt. $\newline$ Calculation: $P(X \geq 3) = 1 - P(X < 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2))$ $\newline$ $P(X = 0) = \binom{20}{0} \cdot (0.30)^0 \cdot (0.70)^{20} = 1 \cdot 1 \cdot 0.70^{20}$ $\newline$ $P(X = 1) = \binom{20}{1} \cdot (0.30)^1 \cdot (0.70)^{19} = 20 \cdot 0.30 \cdot 0.70^{19}$ $\newline$ $P(X = 2) = \binom{20}{2} \cdot (0.30)^2 \cdot (0.70)^{18} = 190 \cdot 0.30^2 \cdot 0.70^{18}$