A process for making plate glass produces small bubbles (imperfections) scattered at random in the glass, at an average rate of 4 small bubbles per #10m^2#. Consider glass pieces with dimensions 2.5m x 2.0m.?
(i) Determine the probability that a piece of glass contains at least one small bubbles.
(ii) Calculate the probability that 5 glass pieces chosen at random are all free of bubbles.
(iii) Suppose 10 glass pieces were randomly chosen. What is the probability that 5 of the pieces contain at least 1 small bubble each?
(i) Determine the probability that a piece of glass contains at least one small bubbles.
(ii) Calculate the probability that 5 glass pieces chosen at random are all free of bubbles.
(iii) Suppose 10 glass pieces were randomly chosen. What is the probability that 5 of the pieces contain at least 1 small bubble each?
1 Answer
(i)
(ii)
(iiI)
Explanation:
(i)
Let
#"Pr"(X=x)" "=(e^(–lambda)lambda^x)/(x!)" "=" "(e^(–2)2^x)/(x!)#
#"Pr"(X>=1)" "=1-"Pr"(X=0)#
#color(white)("Pr"(X>=1))" "=1-"(e^(–2)2^0)/(0!)#
#color(white)("Pr"(X>=1))" "=1-e^(–2)" "~~86.47%#
(ii)
Let
#"Pr"(X_1=0, X_2=0, X_3=0, X_4=0, X_5=0)#
#=prod_(i=1)^5"Pr"(X_i=0)" "# (by independence)
#=["Pr"(X_1=0)]^5" "# (by identical distribution)
#=[(e^(–2)2^0)/(0!)]^5#
#=e^(–10)" "~~0.0045%#
(iii)
Let
#Y" "~" BIN"(n=10," "p=1-e^(–2))#
and
#"Pr"(Y=y)" "=((n),(y))p^y(1-p)^(n-y)#
#color(white)("Pr"(Y=y))" "=((10),(y))(1-e^(–2))^y(e^(–2))^(10-y)#
Then, the probability that exactly 5 pieces contain at least one bubble each is:
#"Pr"(Y=5)" "=((10),(5))(1-e^(–2))^5(e^(–2))^(10-5)#
#color(white)("Pr"(Y=5))" "=252(1-e^(–2))^5e^(–10)#
#color(white)("Pr"(Y=5))" "~~0.553%#
Alternate method for (ii):
The sum of
If
#X_i stackrel "iid"" ~ ""POI"(lambda), i=1,...,k#
then#Y=sum_(i=1)^kX_i" ~ ""POI"(klambda)#
So the chance of 5 pieces of glass containing a total of 0 bubbles across all of them is
#"Pr"(Y=0)" "=" "(e^(–10)10^0)/(0!)#
#color(white)("Pr"(Y=0))" "=" "e^(–10)" "~~0.0045%#
as before.