Conditional probability with given score/outcome

2 Seva has a coin which is biased so that when it is thrown the probability of obtaining a head is \(\frac { 1 } { 3 }\). He also has a bag containing 4 red marbles and 5 blue marbles. Seva throws the coin. If he obtains a head, he selects one marble from the bag at random. If he obtains a tail, he selects two marbles from the bag at random and without replacement.

1. Find the probability that Seva selects at least one red marble.
2. Find the probability that Seva obtains a head given that he selects no red marbles.

7 Susan has a bag of sweets containing 7 chocolates and 5 toffees. Ahmad has a bag of sweets containing 3 chocolates, 4 toffees and 2 boiled sweets. A sweet is taken at random from Susan's bag and put in Ahmad's bag. A sweet is then taken at random from Ahmad's bag.

1. Find the probability that the two sweets taken are a toffee from Susan's bag and a boiled sweet from Ahmad's bag.
2. Given that the sweet taken from Ahmad's bag is a chocolate, find the probability that the sweet taken from Susan's bag was also a chocolate.
3. The random variable \(X\) is the number of times a chocolate is taken. State the possible values of \(X\) and draw up a table to show the probability distribution of \(X\).

3 A fair five-sided spinner has sides numbered 1,2,3,4,5. Raj spins the spinner and throws two fair dice. He calculates his score as follows.

• If the spinner lands on an even-numbered side, Raj multiplies the two numbers showing on the dice to get his score.
• If the spinner lands on an odd-numbered side, Raj adds the numbers showing on the dice to get his score.

Given that Raj's score is 12, find the probability that the spinner landed on an even-numbered side.

3 Jodie tosses a biased coin and throws two fair tetrahedral dice. The probability that the coin shows a head is \(\frac { 1 } { 3 }\). Each of the dice has four faces, numbered \(1,2,3\) and 4 . Jodie's score is calculated from the numbers on the faces that the dice land on, as follows:

• if the coin shows a head, the two numbers from the dice are added together;
• if the coin shows a tail, the two numbers from the dice are multiplied together.

Find the probability that the coin shows a head given that Jodie's score is 8 .

7 Bag \(A\) contains 4 balls numbered 2, 4, 5, 8. Bag \(B\) contains 5 balls numbered 1, 3, 6, 8, 8. Bag \(C\) contains 7 balls numbered \(2,7,8,8,8,8,9\). One ball is selected at random from each bag.

1. Find the probability that exactly two of the selected balls have the same number.
2. Given that exactly two of the selected balls have the same number, find the probability that they are both numbered 2 .
3. Event \(X\) is 'exactly two of the selected balls have the same number'. Event \(Y\) is 'the ball selected from bag \(A\) has number 2'. Showing your working, determine whether events \(X\) and \(Y\) are independent or not.