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The probability that a Kaduna businessman goes to Lagos by car is 0.6, and by air is 0.4. If he goes by car. the probability that he will be on time for his appointment is 0.3 and if he goes by air it is 0.65. (i) Find the probability that he arrives in Lagos early for an appointment. (ii) One Monday, he arrived late for his appointment. Find the probability that he went by car.

Sagot :

Solution:

Given:

[tex]\begin{gathered} P(Car)=0.6 \\ P(Air)=0.4 \end{gathered}[/tex][tex]\begin{gathered} P(Car\text{ and on time\rparen}=0.3 \\ P(air\text{ and on time\rparen}=0.65 \end{gathered}[/tex]

i) The probability that he arrives in Lagos early for the appointment is;

[tex]\begin{gathered} P(early)=(0.3\times0.6)+(0.65\times0.4) \\ P(early)=0.18+0.26 \\ P(early)=0.44 \end{gathered}[/tex]

Therefore, the probability that he arrives in Lagos early for the appointment is 0.44

ii) The probability that he went by car if he arrived late for his appointment is;

To get the probability, we use conditional probability.

[tex]\begin{gathered} P(Car|Late)=\frac{P(car\text{ and late\rparen}}{P(Late)} \\ P(Car|Late)=\frac{P(car)\times P(car\text{ and late\rparen}}{P(late)} \\ P(car\text{ and late\rparen}=1-0.3=0.7 \\ P(car)=0.6 \end{gathered}[/tex]

We now get the probability of late;

[tex]\begin{gathered} P(late)=(0.6\times0.7)+(0.4\times0.35) \\ P(late)=0.42+0.14 \\ P(late)=0.56 \end{gathered}[/tex]

Therefore, the probability that he went by car if he arrived late for his appointment is;

[tex]\begin{gathered} P(Car|Late)=\frac{P(car\text{ and late\rparen}}{P(Late)} \\ P(Car|Late)=\frac{P(car)\times P(car\text{ and late\rparen}}{P(late)} \\ P(car\text{ and late\rparen}=1-0.3=0.7 \\ P(car)=0.6 \\ P(Car|Late)=\frac{0.6\times0.7}{0.56} \\ P(Car|Late)=\frac{0.42}{0.56} \\ P(Car|Late)=0.75 \end{gathered}[/tex]

Therefore, the probability that he went by car if he arrived late for his appointment is 0.75

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