briannaaaa30
23.09.2019 •
Mathematics
An urn contains n white balls andm black balls. (m and n are both positive numbers.) (a) if two balls are drawn without replacement, what is the probability that both balls are the same color? (b) if two balls are drawn with replacement (i.e., one ball is drawn and it’s color recorded and then put back. then the second ball is drawn.) what is the probability that both balls are the same color. (c) show that the probability in part (b) is always larger than the one in part (a).
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Ответ:
DISCLAIMER: Please let me rename b and w the number of black and white balls, for the sake of readability. You can switch the variable names at any time and the ideas won't change a bit!
(a)Case 1: both balls are white.
At the beginning we have balls. We want to pick a white one, so we have a probability of of picking a white one.
If this happens, we're left with white balls and still black balls, for a total of balls. So, now, the probability of picking a white ball is
The probability of the two events happening one after the other is the product of the probabilities, so you pick two whites with probability
Case 2: both balls are black
The exact same logic leads to a probability of
These two events are mutually exclusive (we either pick two whites or two blacks!), so the total probability of picking two balls of the same colour is
(b)Case 1: both balls are white.
In this case, nothing changes between the two picks. So, you have a probability of of picking a white ball with the first pick, and the same probability of picking a white ball with the second pick. Similarly, you have a probability of picking a black ball with both picks.
This leads to an overall probability of
Of picking two balls of the same colour.
(c)We want to prove that
Expading all squares and products, this translates to
As you can see, this inequality comes in the form
With x and y greater than k. This inequality is true whenever the numerator is smaller than the denominator:
And this is our case, because in our case we have
so, y has an extra piece and it is larger which ensures that k<x (and thus k<y), because b and w are integers, and so b<b^2 and w<w^2Ответ:
Find the unit rate: 24 water bottles ÷ 6 dollars total = $0.25 a water bottle
Next, take the rate per water bottle and multiply it by ten: $0.25 a water bottle x 10 bottles of water = $2.50 total
So, this gives you your final answer. Ten bottles of water cost you $2.50!