Suhayb981
08.06.2020 •
Mathematics
Miles draws three cards at random from a standard deck of 52 cards, without replacement.
(a) Find the probability that all three cards are red.
(b) Find the probability that all three cards have the same rank.
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Ответ:
P( red, red, red no replacement ) = 2/17
P( 3 cards with same rank, no replacement) = 1/425
Step-by-step explanation:
There are 52 cards, 26 are red
P( red) = red/total =26/ 52 = 1/2
Now draw the second card without replacement
There are 51 cards, 25 are red
P( red) = red/total =25/51
Now draw the third card without replacement
There are 50 cards, 24 are red
P( red) = red/total =24/50 = 12/25
P( red, red, red no replacement ) = 1/2 * 25/51 * 12/25 = 2/17
P ( all have the same rank)
There are 52 cards, 4 of each rank
We don't care what the first rank is, now the second and third draw have to have the same rank as the first
P( rank) = 1/1
Now draw the second card without replacement
There are 51 cards, 3 are left with the same rank
P( same rank) = cards with same rank/total = 3/51
Now draw the third card without replacement
There are 50 cards, 2 are left with the same rank
P( same rank) = cards with same rank/total = 2/50 = 1/25
P( 3 cards with same rank, no replacement) = 1 * 3/51*1/25 = 1/425
Ответ:
True
False
Step-by-step explanation:
a) The first statement is the basis for Euclid's algorithm to compute the gcd of two nonnegative integers a,b. You can prove this as follows.
Let G=gcd(a,b). Since r=a-bq and G divides a and G divides b, then G divides r. Now, G divides a and G divides r, hence G divides gcd(b,r).
On the other hand, since a=bq+r, and gcd(b,r) divides b and r, then gcd(b,r) divides a. Therefore gcd(b,r) divides a and b, which implies that gcd(b,r) divides G.
x divides y and y divides x implies that |x|=|y|. The GCD's are nonnegative, therefore G=gcd(b,r).
b) It is false. In general, to test for primality of N, you have to check that all primes smaller than N do not divide N. In this case, we have to check for 2,3,5,7,11,13,17,19,23,...
101 is prime, but this may be false in general. For example, consider N=13*11=143. N is not prime, and n is not divisible by 2,3,5, or 7.