Ten balls of identical size are in a bag. Two of the balls are red and the rest are blue. Balls are picked out at random from the bag and are not replaced. Let $R$ be the number of balls drawn out, up to and including the first red one. List the possible values of $R$ and their associated probabilities.

Ten balls of identical size are in a bag. Two of the balls are red and the rest are blue. Balls are picked out at random from the bag and are not replaced. Let $R$ be the number of balls drawn out, up to and including the first red one. Calculate the mean value of $R$.

Ten balls of identical size are in a bag. Two of the balls are red and the rest are blue. Balls are picked out at random from the bag and are not replaced. Let $R$ be the number of balls drawn out, up to and including the first red one. What is the most likely value of $R$ ?

Consider the bag of balls in question $8$ Suppose now that each ball is replaced before the next is drawn. Show that the probability of the first red being drawn out on the second go is $\frac{4}{25}$.

Consider the bag of balls in question $8$. Suppose now that each ball is replaced before the next is drawn. Calculate the probability that the first red is drawn after the third go.

7 Consider the bag of balls in question 8 Suppose now that each ball is replaced before the next is drawn. Derive a formula to find the probability that the first red is drawn on the $n^{th}$ go.

Consider the bag of balls in question $8$ Suppose now that each ball is replaced before the next is drawn. What is the most likely value of $R$ ?

Suppose an instant lottery ticket is purchased for $\$2$. The possible prizes are $\$0, \$2, \$20, \$200$ and $\$1000$. Let $Z$ be the random variable representing the amount won on the ticket, and suppose $Z$ has the following distribution.

Determine $P(Z=0)$.

Suppose an instant lottery ticket is purchased for $\$2$. The possible prizes are $\$0, \$2, \$20, \$200$ and $\$1000$. Let $Z$ be the random variable representing the amount won on the ticket, and suppose $Z$ has the following distribution.

Determine $E\left(Z\right)$ and interpret its meaning.