Question 1032824
A = event the card is a heart
B = event the card is a red card 
Which of the following is (are) true?
A: P(A or B) = P(A)
B: P(A or B) = P(B)
C: P(A and B) = P(B)
D: P(A and B) = P(A) 
<pre>
Here is the universal set of all 52 cards in the deck:
cards 
<font color = "red">
 
A&#9829;   2&#9829;   3&#9829;   4&#9829;   5&#9829;   6&#9829;   7&#9829;   8&#9829;  9&#9829;  10&#9829;  J&#9829;  Q&#9829;  K&#9829; 
A&#9830;   2&#9830;   3&#9830;   4&#9830;   5&#9830;   6&#9830;   7&#9830;   8&#9830;  9&#9830;  10&#9830;  J&#9830;  Q&#9830;  K&#9830;</font>
A&#9824;   2&#9824;   3&#9824;   4&#9824;   5&#9824;   6&#9824;   7&#9824;   8&#9824;  9&#9824;  10&#9824;  J&#9824;  Q&#9824;  K&#9824;  
A&#9827;   2&#9827;   3&#9827;   4&#9827;   5&#9827;   6&#9827;   7&#9827;   8&#9827;  9&#9827;  10&#9827;  J&#9827;  Q&#9827;  K&#9827; 

A = event the card is a heart

Here is subset A

<font color = "red">
 
A&#9829;   2&#9829;   3&#9829;   4&#9829;   5&#9829;   6&#9829;   7&#9829;   8&#9829;  9&#9829;  10&#9829;  J&#9829;  Q&#9829;  K&#9829;</font>

There are 13 of them out of the 52 cards, therefore

P(A) = 13/52 = 1/4

B = event the card is a red card

Here is subset B
<font color = "red">
A&#9829;   2&#9829;   3&#9829;   4&#9829;   5&#9829;   6&#9829;   7&#9829;   8&#9829;  9&#9829;  10&#9829;  J&#9829;  Q&#9829;  K&#9829; 
A&#9830;   2&#9830;   3&#9830;   4&#9830;   5&#9830;   6&#9830;   7&#9830;   8&#9830;  9&#9830;  10&#9830;  J&#9830;  Q&#9830;  K&#9830;</font>

There are 26 of them out of the 52 cards, therefore

P(B) = 26/52 = 1/2.

P(A or B)

That's the probability that a card is a heart or a red card,
which just means that it is a red card, since hearts are red.

Here is the subset A or B:
<font color = "red">
A&#9829;   2&#9829;   3&#9829;   4&#9829;   5&#9829;   6&#9829;   7&#9829;   8&#9829;  9&#9829;  10&#9829;  J&#9829;  Q&#9829;  K&#9829; 
A&#9830;   2&#9830;   3&#9830;   4&#9830;   5&#9830;   6&#9830;   7&#9830;   8&#9830;  9&#9830;  10&#9830;  J&#9830;  Q&#9830;  K&#9830;</font>

There are 26 of them out of the 52 cards, therefore

P(A or B) = 26/52 = 1/2  

P(A or B) is the same as P(B).  Therefore B is true.
------------

P(A and B)

That's the probability that a card is a heart and a red card,
which just means that it is a heart, since hearts are red.

Here is subset A and B:

<font color = "red">
 
A&#9829;   2&#9829;   3&#9829;   4&#9829;   5&#9829;   6&#9829;   7&#9829;   8&#9829;  9&#9829;  10&#9829;  J&#9829;  Q&#9829;  K&#9829;</font>

There are 13 of them out of the 52 cards, therefore

P(A and B) = 13/52 = 1/4  

P(A and B) is the same as P(A).  Therefore D is true.

So B and D are true and A and C are false.

Edwin</pre>