Use the table to calculate the probability that it is a Thursday, given that Jemma has travelled to work either by a train or a bus? The table show the relative frequencies with which Jemma uses each type of transport on her two travel day.

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2 Answers
Jan 26, 2018

#P("Thursday"|"Train or bus")=0.5#

Explanation:

The probability is written as #P("Thursday"|"Train or bus")#, or "The probability it is a Thursday, given a train or bus was taken".

To fond this probability, we find the probability that either a bus or train was taken and divide it by the probability a bus or train was taken.

#P("Thursday"|"Train or bus")=(P("Train or bus on Thursday"))/(P("Train or bus was taken"))#

#P("Train or bus on Thursday")=0.18+0.17=0.35#

#P("Train or bus was taken")=0.18+0.17+0.15+0.20=0.70#

#P("Thursday"|"Train or bus")=0.35/0.70=1/2=0.5#

Jan 26, 2018

#0.50#

Explanation:

Note that #"Train"# / #"Car"# / #"Bus"# are mutually exclusive, as are #"Monday"# / #"Thursday"#.

For brevity, let us write:

#{ (T = "Train"), (C = "Car"), (B = "Bus"), ("Mo" = "Monday"), ("Th" = "Thursday") :}#

Then:

#P("Th" | (T uu B)) = (P("Th" nn (T uu B)))/(P(T uu B))#

Now:

#P("Th" nn (T uu B)) = P("Th" nn T) + P("Th" nn B) = 0.18 + 0.17 = 0.35#

and:

#P(T uu B) = P(("Mo" uu "Th") nn (T uu B))#

#color(white)(P(T uu B)) = P(("Mo" nn T) uu ("Mo" nn B) uu ("Th" nn T) uu ("Th" nn B))#

#color(white)(P(T uu B)) = P("Mo" nn T) + P("Mo" nn B) + P("Th" nn T) + P("Th" nn B)#

#color(white)(P(T uu B)) = 0.15+0.20+0.18+0.17 = 0.70#

So:

#P("Th" | (T uu B)) = (P("Th" nn (T uu B)))/(P(T uu B)) = 0.35/0.70 = 0.50#