Three cards are drawn in succession, without replacement, from an ordinary deck of 52 cards, Find the probability that the first card is a red ace, the second card is a 10 or Jack and the third card is greater than 3 but less than 7.

1. Three cards are drawn in succession, without replacement, from an
ordinary deck of 52 cards, Find the probability that the first card is a
red ace, the second card is a 10 or Jack and the third card is greater
than 3 but less than 7.
Solution
There are 2 red Ace card in a deck, therefore
𝑷(𝑭𝒊𝒓𝒔𝒕 𝒄𝒂𝒓𝒅 𝒊𝒔 𝒂 𝒓𝒆𝒅 𝑨𝒄𝒆) = 𝑷(𝑨) =
𝟐
𝟓𝟐
There are 4 cards of 10 and 4 cards of Jack in a deck, second card
will be drawn from the remaining 51 cards. Therefore
𝑷(𝑺𝒆𝒄𝒐𝒏𝒅 𝒄𝒂𝒓𝒅 𝒊𝒔 𝒂 𝟏𝟎 𝒐𝒓 𝑱𝒂𝒄𝒌) = 𝑷(𝑩) =
𝟖
𝟓𝟏
There exists 12 card which are greaterthan 3 but less than 7, therefore
𝑷(𝑻𝒉𝒊𝒓𝒅 𝒄𝒂𝒓𝒅 𝒊𝒔 𝐠𝐫𝐞𝐚𝐭𝐞𝐫 𝐭𝐡𝐚𝐧 𝟑 𝐛𝐮𝐭 𝐥𝐞𝐬𝐬 𝐭𝐡𝐚𝐧 𝟕) = 𝑷(𝑪) =
𝟏𝟐
𝟓𝟎
Probability that three cards are drawn in succession, without replacement is
𝑷(𝑨 ∩ 𝑩 ∩ 𝑪) = 𝑷(𝑨) × 𝑷(𝑩
𝑨
⁄ ) × 𝑷(𝑪
𝑨 ∩ 𝑩
⁄ )
𝑷(𝑨 ∩ 𝑩 ∩ 𝑪) =
𝟐
𝟓𝟐
×
𝟖
𝟓𝟏
×
𝟏𝟐
𝟓𝟎
= 𝟎. 𝟎𝟎𝟏𝟒𝟓