HOW CAN YOU USE DISCRETE PROBABILITY TO CALCULATE THE PROBABILITY OF TESTING 5 DEVICES SUCH THAT ALL 5 END UP BEING NONDEFECTIVE, GIVEN THERE ARE 20 DEFECTIVE DEVICES TOTAL OF THE ORIGINAL 100 DEVICES? WHAT IS THE SAMPLE SPACE?
p fail = .2
p pass = .8
p five pass = .8^5 = .32768
The finite population has 100 devices, out of which 80 are non-defective.
At each draw, both the total number devices and the number of non-defective device decrease, we have the running probabilities as:
{80/100, 79/99, 78/98, 77/97, 76/96}
the product of which give 0.31931, very close the the earlier estimate of 0.32.
Alternatively, the hypergeometric distribution may be used, which gives
P(5 nondefectives)
=C(20,0)*C(80,5)/C(100,5)
=0.31931 as before.
To find Ω, we determine that there 2 possible outcomes for each draw, hence for 5 draws, the sample space is 2^5=32.
To calculate the probability of testing 5 devices and all of them being non-defective, we can use discrete probability.
First, we need to determine the sample space, which represents all possible outcomes of the testing. Since there are 100 devices total and 20 of them are defective, the sample space will consist of all possible combinations of selecting 5 devices out of the 100.
The total number of combinations can be calculated using the concept of combinations. The formula for combinations is:
C(n, r) = n! / (r!(n-r)!)
where n is the total number of devices (100 in this case), and r is the number of devices tested (5 in this case). By plugging in these values, we can calculate the total number of combinations:
C(100, 5) = 100! / (5!(100-5)!)
= 100! / (5!95!)
Calculating this value will give us the total number of combinations in the sample space.
Next, we need to determine the favorable outcomes, which are the combinations where all 5 tested devices are non-defective. Since there are 20 defective devices in total, there are 80 non-defective devices. We need to select all 5 devices from this pool of 80 non-defective devices.
The number of combinations of selecting 5 non-defective devices from a pool of 80 can be calculated using the same combination formula:
C(80, 5) = 80! / (5!(80-5)!)
= 80! / (5!75!)
Calculating this value will give us the number of favorable outcomes.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of combinations:
Probability = Number of favorable outcomes / Total number of combinations
In this case, the probability will be:
Probability = C(80, 5) / C(100, 5)
= (80! / (5!75!)) / (100! / (5!95!))
By simplifying this expression, we can obtain the final probability.