Suppose that an ordinary deck of $52$cards is shuffled and the cards are then turned over one at a time until the first ace appears. Given that the first ace is the role="math" localid="1647784076635" ${20}^{th}$card to appear, what is the conditional probability that the card following it is the

1. ace of spades?
2. two of the clubs?

Given in the question that the first ace is the ${20}^{th}$card to appear and the conditional probability that the card following it is the ace of spades.

The probability that ${20}^{\text{th}}$card is not Ace of spade $=\frac{3}{4}$

From 21 to$52=(32cards)$

In 32 cards there is one Ace of spade $\mathrm{p}=\frac{1}{32}$

The probability that role="math" localid="1647784603931" ${21}^{\mathrm{st}}$card is Ace of spade $=\frac{3}{4}\times \frac{1}{32}$

$=\frac{3}{128}$

The probability card ${21}^{\text{st}}$is Ace of spade$\frac{3}{128}.$

Given in the question that the first ace is the ${20}^{th}$card to appear and the conditional probability that the card following it is the two of clubs.

The first $19$cards can be any of $48$the cards (excluding

From $21$to $52$($32$cards)

There are $3$Ace's and $2$$\mathrm{P}\left({21}^{\text{st}}\right.$card is two of clubs)

$\frac{29}{1536}$.

The conditional probability that the card is the two of clubs is $\frac{29}{1536}.$