Finding unknown probability from total probability

3 On each day that Alexa goes to work, the probabilities that she travels by bus, by train or by car are \(0.4,0.35\) and 0.25 respectively. When she travels by bus, the probability that she arrives late is 0.55 . When she travels by train, the probability that she arrives late is 0.7 . When she travels by car, the probability that she arrives late is \(x\). On a randomly chosen day when Alexa goes to work, the probability that she does not arrive late is 0.48 .

1. Find the value of \(x\).
2. Find the probability that Alexa travels to work by train given that she arrives late.

2 The probability that a student at a large music college plays in the band is 0.6. For a student who plays in the band, the probability that she also sings in the choir is 0.3 . For a student who does not play in the band, the probability that she sings in the choir is \(x\). The probability that a randomly chosen student from the college does not sing in the choir is 0.58 .

1. Find the value of \(x\).
   Two students from the college are chosen at random.
2. Find the probability that both students play in the band and both sing in the choir.

1 On any day, Kino travels to school by bus, by car or on foot with probabilities 0.2, 0.1 and 0.7 respectively. The probability that he is late when he travels by bus is \(x\). The probability that he is late when he travels by car is \(2 x\) and the probability that he is late when he travels on foot is 0.25 . The probability that, on a randomly chosen day, Kino is late is 0.235 .

1. Find the value of \(x\).
2. Find the probability that, on a randomly chosen day, Kino travels to school by car given that he is not late.

2 Maria has 3 pre-set stations on her radio. When she switches her radio on, there is a probability of 0.3 that it will be set to station 1, a probability of 0.45 that it will be set to station 2 and a probability of 0.25 that it will be set to station 3 . On station 1 the probability that the presenter is male is 0.1 , on station 2 the probability that the presenter is male is 0.85 and on station 3 the probability that the presenter is male is \(p\). When Maria switches on the radio, the probability that it is set to station 3 and the presenter is male is 0.075 .

1. Show that the value of \(p\) is 0.3 .
2. Given that Maria switches on and hears a male presenter, find the probability that the radio was set to station 2.

1 When Shona goes to college she either catches the bus with probability 0.8 or she cycles with probability 0.2 . If she catches the bus, the probability that she is late is 0.4 . If she cycles, the probability that she is late is \(x\). The probability that Shona is not late for college on a randomly chosen day is 0.63 . Find the value of \(x\).

2 Benju cycles to work each morning and he has two possible routes. He chooses the hilly route with probability 0.4 and the busy route with probability 0.6 . If he chooses the hilly route, the probability that he will be late for work is \(x\) and if he chooses the busy route the probability that he will be late for work is \(2 x\). The probability that Benju is late for work on any day is 0.36 .

1. Show that \(x = 0.225\).
2. Given that Benju is not late for work, find the probability that he chooses the hilly route.

1. A factory buys \(10 \%\) of its components from supplier \(A , 30 \%\) from supplier \(B\) and the rest from supplier \(C\). It is known that \(6 \%\) of the components it buys are faulty.

Of the components bought from supplier \(A , 9 \%\) are faulty and of the components bought from supplier \(B , 3 \%\) are faulty.

1. Find the percentage of components bought from supplier \(C\) that are faulty. A component is selected at random.
2. Explain why the event "the component was bought from supplier \(B\) " is not statistically independent from the event "the component is faulty".

Nikita goes shopping to buy a birthday present for her mother. She buys either a scarf, with probability 0.3, or a handbag. The probability that her mother will like the choice of scarf is 0.72. The probability that her mother will like the choice of handbag is \(x\). This information is shown on the tree diagram. The probability that Nikita's mother likes the present that Nikita buys is 0.783.

1. Find \(x\). [3]
2. Given that Nikita's mother does not like her present, find the probability that the present is a scarf. [4]

A sample of 200 households was obtained from a small town. Each household was asked to complete a questionnaire about their purchases of takeaway food. \(A\) is the event that a household regularly purchases Indian takeaway food. \(B\) is the event that a household regularly purchases Chinese takeaway food. It was observed that \(P(B|A) = 0.25\) and \(P(A|B) = 0.1\) Of these households, 122 indicated that they did not regularly purchase Indian or Chinese takeaway food. A household is selected at random from those in the sample. Find the probability that the household regularly purchases both Indian and Chinese takeaway food. [6 marks]

A factory buys 10\% of its components from supplier \(A\), 30\% from supplier \(B\) and the rest from supplier \(C\). It is known that 6\% of the components it buys are faulty. Of the components bought from supplier \(A\), 9\% are faulty and of the components bought from supplier \(B\), 3\% are faulty.

1. Find the percentage of components bought from supplier \(C\) that are faulty. [3]

A component is selected at random.

1. Explain why the event "the component was bought from supplier \(B\)" is not statistically independent from the event "the component is faulty". [1]