Question

The probability of a bank making a mistake in processing a deposit is 0.0003. If 10000 deposits are audited, what is the probability that more than 6 mistakes were made in processing deposits?

Use Poisson distribution. Use (lambda formula)

Answer 1

Given:

The probability of a bank making a mistake in processing a deposit is 0.0003.

To Find:

If 10000 deposits are audited, what is the probability that more than 6 mistakes were made in processing deposits?

Solution:

For this sum, we will have to use the Poisson distribution and for that, we need to calculate the lamba. so,

                                              $\lambda=\mu=n*p\\=10000*0.0003\\=3$

Because n>20 and $n*p\leq 7$, the Poisson approximation is close enough to analyze x>6. so now we need to find Poisson distribution using the formula for x>6. the formula for the same is

                                            $P(x)=\frac{\lambda^{x}e^{-\lambda} }{x!}$

So, for x=7

                                            $P(7)=\frac{3^{7}*e^{-3} }{7!}\\=0.0216$

Similarly, we need to calculate till x=12

P(7)=0.0216

P(8)=0.0081

P(9)=0.0027

P(10)=0.0008

P(11)=0.0002

P(12)=0.0001

Now add up all the values we get,$P=P(7)+P(8)+P(9)+P(10)+P(11)+P(12)\\=0.0216+0.0081+0.0027+0.0008+0.0002+0.0001\\=0.0335$

Hence, the probability that more than 6 mistakes were made in processing deposits is 0.0335.