We have been given that on a coordinate plane a line with negative slope passes through points and
. We are asked to find the rate of the change of the function.
We know that slope represents rate of change of function.
We will use slope formula to solve our given problem.
Let and
.
Therefore, the rate of change of the given function is .
In a hand of 5 cards, you want 4 of them to be of the same rank, and the fifth can be any of the remaining 48 cards. So if the rank of the 4-of-a-kind is fixed, there are possible hands. To account for any choice of rank, we choose 1 of the 13 possible ranks and multiply this count by
. So there are 624 possible hands containing a 4-of-a-kind. Hence A occurs with probability
There are 4 aces in the deck. If exactly 1 occurs in the hand, the remaining 4 cards can be any of the remaining 48 non-ace cards, contributing possible hands. Exactly 2 aces are drawn in
hands. And so on. This gives a total of
possible hands containing at least 1 ace, and hence B occurs with probability
The product of these probability is approximately 0.000082.
A and B are independent if the probability of both events occurring simultaneously is the same as the above probability, i.e. . This happens if
The above "sub-events" are mutually exclusive and share no overlap. There are 48 possible non-aces to choose from, so the first sub-event consists of 48 possible hands. There are 12 non-ace 4-of-a-kinds and 4 choices of ace for the fifth card, so the second sub-event has a total of 12*4 = 48 possible hands. So consists of 96 possible hands, which occurs with probability
and so the events A and B are NOT independent.