Question

A scientist has three batches of blood samples available on which to run some tests. Each batch contains exactly 8 blood samples and each blood sample is either a normal or a viral sample. In the first batch 6 of the blood samples are viral, in the second batch 4 of the blood samples are viral and in the third batch 3 of the blood samples are viral. The scientist will first choose a batch, then select two blood samples (without replacement) from that batch. The probabilities of choosing the first, second and third batches are 0.57, 0.11 and 0.32, respectively. Find the probability that both selected blood samples are viral.

Answer 1

Define Bk= k th batch is selected , k=1,2,3

A=Both selected samples are viral

P(A)=P(B1)P(A|B1)+P(B2)P(A|B2​​​​​​)+P(B3)P(A|B3)

Now P(B1)=.57,P(B2)=.11, P(B3)=.32​​

P(A|B1)= =15/28 as in the first batch there are 6 viral samples and to get two viral samples, we need to take 2 samples from those 6. Also 2 out of 8 can be chosen in ways.

P(A|B2)= = 3/14 as in the second  batch there are 4  viral samples and to get two viral samples, we need to take 2 samples from those 4. Also 2 out of 8 can be chosen in ways.

P(A|B3)= =3/28 as in the first batch there are 3 viral samples and to get two viral samples, we need to take 2 samples from those 3. Also 2 out of 8 can be chosen in ways.

Then P(A)=.57*(15/28)+.11*(3/14)+.32*(3/28)=0.3632143 is the required probability.

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