Martin S. answered • 04/10/20
Patient, Relaxed PhD Molecular Biologist for Science and Math Tutoring
Since Aces are considered as face cards for this problem, there are 4 faces cards (Jack, Queen, King and Ace). Each of the four suits have four face cards, so a total of 16 face cards in the deck
a. The card is black, and it is a face card. P(black) = 1/2, and P(face card) = 16/52 = 4/13. Multiply the two probabilities as they are independent and must both occur. P(black AND face card) = 1/2 x 4/13 = 2/26 = 1/3.
b. P(black OR face card) For this, as either of two conditions can occur, add probabilities. But be careful not to count any card twice. Keep in mind that some face cards are black, and must only be counted once. So we can add the probability of the card being black,and the probability of the card being a red face card (remember, the black face cards are already accounted for in the group of black cards. P(black) is simply 1/2. P(red AND face card) works out to the same as the previous problem, P(red) = 1/2, and P(face card) = 4/13. Multiply those probabilities and P(red And face card) = 1/13. Now add the two probabilities, 1/2 + 1/13 = 13/26 + 2/25 = 15/26.
c. P(black OR 3). Similar to problem b, but there are fewer red cards to consider (only the two red 3's instead of the 8 red face cards). Again, determine probabilities of subgroups and add them, being careful to only count each card once. P(black) = 1/2; P(red AND 3) = P(red) x (P3) = 1/2 x 1/3 = 1/26. Now add the probabilities of the two subgroups, P(black) + P(red AND 3) = 1/2 + 1/26 = 13/26 + 1/26 =14/26 =7/13.
d. P(face card OR number card). Since each card in the deck must be either a face card or a number card (there are no Jokers in this deck), all the cards are accounted for and the probability must be equal to 1. But let's check that mathematically. We have two independent conditions, and either one can satisfy the statement. So we determine the probabilities and add. There are 16 face cards, so P(face card) = 16/52 = 4/13. There are 9 number values and four suits, so there are 36 number cards. P(number card) then is 36/52 = 9/13. Add the two probabilities and 4/13 +9/13 = 13/13 = 1. A nice demonstration of the basic theorem, p + q = 1.
e. P(black AND queen). These are two conditions that must occur simultaneously, so multiply the two individual probabilities. P(black) = 1/2, and P(queen) = 4/52 = 1/13. Multiply and 1/2 x 1/13 = 1/26.
Hope this helps
