Question

A junk box in your room contains fourteen old​ batteries, seven of which are totally dead. You start picking batteries one at a time and testing them. Find the probability of each outcome. ​a) The first two you choose are both good. ​b) At least one of the first three works. ​c) The first four you pick all work. ​d) You have to pick five batteries to find one that works.

Answer 1

Answer:

A junk box in your room contains fourteen old​ batteries, seven of which are totally dead.

So, number of good batteries = $14-7=7$

a) The first two you choose are both good. ​

There is a 7/14 chance that we will pick a good battery.

Now there are 13 batteries left and out of that there are only 6 good ones remaining, so this becomes 6/13.

So, combined probability is = $\frac{7}{14}\times \frac{6}{13}$

= 0.23

b) At least one of the first three works.

$\frac{7}{14} \times \frac{6}{13} \times \frac{5}{12} =0.096$

And at least one is good battery, we get : $1-0.096=0.904$

​c) The first four you pick all work.

There is a probability of 7/14 for the first one to work, 6/13 for the second, 5/12 for the third and 4/11 when the fourth is good, combined we get by multiplying all:

$\frac{7}{14}\times \frac{6}{13}\times \frac{5}{12}\times \frac{4}{11}=0.0344$

d) You have to pick five batteries to find one that works.

This condition means that we pick 4 bad batteries and 1 good battery.

The probability of picking 4 bad batteries is -

$\frac{7}{14}\times \frac{6}{13}\times \frac{5}{12}\times \frac{4}{11}=0.0344$

Since there are 7 good batteries remaining in 10 batteries so we will multiply 7/10 in 0.0344 to know the fifth one that finally works.  

This becomes = $0.0344\times0.7=0.024$