The total number of cards in a deck of Italian playing cards is 40. Out of those 40 playing cards there are ten cards that are Cups. Consider an experiment where the deck is properly shuffled and a single card is dealt.

(a) What is the probability that the card dealt is a Cup?

(b) Now suppose that the card is returned to the deck and that the deck is once again properly shuffled. The experiment is repeated another 47 times (i.e. in total 48 trials of the experiment are conducted - including the initial trial in (a) above) where the card that is dealt is returned to the deck after each trial and the deck is properly shuffled again before the next card is dealt. Out of the 48 trials, let X equal the number of Cup cards that are observed.

i. What are all of the possible values that can be observed for X?

ii. What is the expected frequency for the number of Cup cards dealt? Show workings.

iii. Suppose that X = 16 is observed. What is the relative frequency for this many Cup cards dealt?

iv. Now suppose that the experiment is to be repeated but this time with a very large n (number of trials). For a very large n, what should the relative frequency for the number of Cup cards dealt be approximately equal to?