# Probability question on drawing cards

#### micky123

##### New member
From a standard pack of 52 cards, I draw 3 cards at random without replacement. What is the probability that all cards drawn are both hearts and picture cards?

#### Subhotosh Khan

##### Super Moderator
Staff member
From a standard pack of 52 cards, I draw 3 cards at random without replacement. What is the probability that all cards drawn are both hearts and picture cards?

There are 4 cards (ACE included) with pictures AND heart.

How many ways can you draw 3 cards from that group ?

How many ways can you draw 3 cards from 52 cards ?

#### micky123

##### New member
Why would ace be included? I thought there are only 3 cards that are both heart and picture? (jack, queen and king)

Please kindly explain if I have gotten it wrong! Thank you.

#### Subhotosh Khan

##### Super Moderator
Staff member
Why would ace be included? I thought there are only 3 cards that are both heart and picture? (jack, queen and king)

Please kindly explain if I have gotten it wrong! Thank you.
If ACE is not included then - how many ways can you choose 3 cards from that group (Hearts and pictures) ? ------ 1 way

How many ways can you choose 3 cards (no restriction) from the deck? _____ $$\displaystyle C^{52}_{3}$$

Now calculate probability.

reference: https://math.stackexchange.com/ques...-getting-3-cards-in-the-same-suit-from-a-deck

#### micky123

##### New member
If ace is not included, then the probability is 1/22100. However, i am confused as to why there are 4 cards with pictures and heart?

Am I understanding the event incorrectly? I understood the event that “all three cards are both heart and picture” to be drawing “a jack, a queen and a king of hearts”, which is 1 out of 22100 ways.

#### pka

##### Elite Member
If ace is not included, then the probability is 1/22100. However, i am confused as to why there are 4 cards with pictures and heart?

Am I understanding the event incorrectly? I understood the event that “all three cards are both heart and picture” to be drawing “a jack, a queen and a king of hearts”, which is 1 out of 22100 ways.
BUT if I were reading this question as an editor I would object to its wording as being unclear. It is extremely unusual to use picture card what standard English uses face cards.

#### mmm4444bot

##### Super Moderator
Staff member
It is extremely unusual to use picture card what standard English uses face cards.
Your wikipedia link states that names 'picture card' and 'face card' mean the same thing.

#### Otis

##### Elite Member
confused as to why there are 4 cards with pictures and heart?
Because a lot of card manufacturers draw a picture on each ace. However, the design on the aces do not show a person.

The name 'picture card' refers only to cards containing a picture of a person.

#### micky123

##### New member
I see. So this means that it depends on the way “picture card” is defined in the question?

if ace is not considered a picture card, the ans would be (3/52)*(2/51)*(1/50).

if ace is considered a picture card, the and would be (4/51)*(3/51)*(2/50).

am i right?