Question

The number of contaminating particles on a silicon wafer prior to a certain rinsing process was determined for each wafer in a sample of size 100, resulting in the following frequencies:Number of particles: 0, 1, 2, 3, 4, , 5, 6, 7Frequency: 1, 2, 3, 12, 11, 15, 18, 10Number of particles: 8, 9, 10, 11, 12, 13, 14Frequency: 12, 4, 5, 3, 1, 2, 1What proportion of the sampled wafers had at least one particle? At least five particles?

Answer 1

Answer:

a) $P(X\geq 1) = 1-P(X<1) = 1-P(X=0) =1-0.01= 0.99$

b) $P(X\geq 5) = 1-P(X<5) = 1-P(X\leq 4) = 1-[P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)]$

And replacing we got:

$P(X \geq 5) = 1-[0.01+0.02+0.03+0.12+0.11]=1-0.29=0.71$

Step-by-step explanation:

For this case we can solve this problem creating the following table

Number of particles      Frequency         Rel. Frequency

             0                            1                       1/100 =0.01

             1                             2                      2/100 =0.02

             2                            3                       3/100=0.03

             3                            12                     12/100=0.12    

             4                            11                       11/100=0.11

             5                            15                      15/100=0.15

             6                            18                      18/100=0.18

             7                            10                       10/100=0.1

             8                            12                       12/100=0.12

             9                            4                         4/100=0.04

             10                           5                         5/100=0.05

             11                            3                         3/100=0.03

             12                           1                          1/100=0.01

             13                           2                         2/100=0.02

             14                           1                          1/100=0.01

           Total                       100                            1

We assume on this case the the relative frequency represent the probability.

Let X the number of contaminating particles on a silicon wafer

What proportion of the sampled wafers had at least one particle?

For this case we can use the complement rule like this:

$P(X\geq 1) = 1-P(X<1) = 1-P(X=0) =1-0.01= 0.99$

At least five particles?

Again for this case we can use the complement rule and we got:

$P(X\geq 5) = 1-P(X<5) = 1-P(X\leq 4) = 1-[P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)]$

And replacing we got:

$P(X \geq 5) = 1-[0.01+0.02+0.03+0.12+0.11]=1-0.29=0.71$