Question

Find the probability of randomly selecting

44

good quarts of milk in succession from a cooler containing

2323

quarts of which

55

have spoiled. Complete parts​ (a) and​ (b) below.

​(a) Use the following​ formula, where

Upper A 1A1​,

Upper A 2A2​,...,Upper A Subscript kAk

are events that can occur in an experiment.

Upper P left parenthesis Upper A 1 intersect Upper A 2 intersect ... intersect Upper A Subscript k Baseline right parenthesisPA1 ∩ A2 ∩ ... ∩ Ak

equals Upper P left parenthesis Upper A 1 right parenthesis Upper P left parenthesis Upper A 2 | Upper A 1 right parenthesis Upper P left parenthesis Upper A 3 | Upper A 1 intersect Upper A 2 right parenthesis times times times Upper P left parenthesis Upper A Subscript k Baseline | Upper A 1 intersect Upper A 2 intersect times times times intersect Upper A Subscript k minus 1 Baseline right parenthesis=PA1PA2 | A1PA3 | A1 ∩ A2•••PAk | A1 ∩ A2 ∩•••∩ Ak−1

equals=nothing

​(Do not perform the calculation. Use integers or fractions for any numbers i

Answer 1

Total number of quarts = 23

Number of spoiled quarts = 5

Number of good quarts =23 -5 =18

Formula used:

$P(A1\bigcap A2\bigcap ...\bigcap Ak)=P(A1)+(A2/A1)+(A3/A1\bigcap A2)+...P(Ak/A1\bigcap A2\bigcap Ak-1)$

Here:

k = 4

Substituting, we have:

$P(A1\bigcap A2\bigcap A3 \bigcap A4)=P(A1)+(A2/A1)+(A3/A1\bigcap A2)+P(A4/A1\bigcap A2\bigcap A3)$

$P(A1) = 18/23$

$P(A2/A1) = 17/22$

$P(A3/A1\bigcap A2) = 16/21$

$P(A4/A1\bigcap A2\bigcap A3) = 15/20$

Thus,the answer is:

$(\frac{18}{23})\times (\frac{17}{22})\times (\frac{16}{21})\times (\frac{15}{20})$