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 = $0.0344\times0.7=0.024$

Answer 2

The various probabilities of each of the given outcomes for selection of a battery from the junk box are; a) 0.23 b) 0.904 c) 0.0344 d) 0.0241

What is the probability outcome?

We are told that the junk box contains fourteen old​ batteries, seven of which are totally dead. Thus;

Number of good batteries = 7

Number of dead batteries = 7

a) We want to find the probability that the first two you choose are both good. ​

P(first is good) = 7/14

P(second picked is good) = 6/13

Thus;

P(both first and second picked are good) = (7/14) * (6/13)

P(both first and second picked are good) = 0.23

b)  We want to find the probability that least one of the first three works. Thus;

P(first works) = 7/14

P(second picked works) = 6/13

P(third picked works) = 5/12

P(first 3 works) = (7/14) * (6/13) * (5/12)

P(first 3 works) = 0.096

P(at least one of the first 3 works) = 1 - 0.096

P(at least one of the first 3 works) = 0.904

c) P(first four all work) =  (7/14) * (6/13) * (5/12) * (4/11)

P(first four all work) = 0.0344

d) Probability of first four being bad batteries is;

P(first four not working) =  (7/14) * (6/13) * (5/12) * (4/11)

P(first four not working) = 0.0344

Probability of the fifth one working = 7/10 = 0.7

Thus;

P(pick 5 before seeing one that works) = 0.0344 * 0.7

P(pick 5 before seeing one that works) = 0.0241

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