Suppose that an ordinary deck of 52 cards (which contains 4 aces) is randomly divided into 4 hands of 13 cards each. We are interested in determining p, the probability that each hand has an ace. Let
Ei
be the event that the ith hand has exactly one ace. Determine
p=P(E1E2E3E4)
by using the multiplication rule.

P ( E_1*E_2*E_3*E_4 ) = 0.1055

Step-by-step explanation:

Given:

- 52 cards are dealt in 1 , 2 , 3 , 4 hands.

- Events:

             E_1       Hand 1 has exactly 1 ace

             E_2       Hand 2 has exactly 1 ace

             E_3       Hand 3 has exactly 1 ace

             E_4       Hand 4 has exactly 1 ace

Find:

p =P ( E_1*E_2*E_3*E_4 )

Solution:

Multiplication rule.

- For n ε N and events E_1 , E_2 , ... , E_n:

P ( E_1*E_2**E_n ) = P (E_1)*P(E_2|E_1)*P(E_3|E_2*E_1)**(E_n|E_1*E_2...E_n-1 )

- So for these events calculate 4 probabilities:-

-  For E_1, is to choose an ace multiplied by the number of ways to choose remaining 12 cards out of 48 non-aces:

                               P ( E_1 ) = 4C1 * 48C12 / 52C13

- For E_2 | E_1 , one ace and 12 other cards have already been chosen. there are 39C13 equally likely hands. The number of different one ace hand 2 is the number of ways to choose an ace from 3 remaining multiplied by the number of ways to choose the remaining 12 from 36, we have:

                               P ( E_2 | E_1  ) = 3C1 * 36C12 / 39C13

                               P ( E_3| E_2*E_1  ) = 2C1 * 24C12 / 26C13

                               P ( E_4 | E_3*E_2*E_1  ) = 1C1*12C12 / 13C13 = 1

- So the multiplication rule for n = 4 is as follows:

     P ( E_1*E_2*E_3*E_4 ) = P (E_1)*P(E_2|E_1)*P(E_3|E_2*E_1)*P ( E_4 | E_3*E_2*E_1  ) = [ 4C1 * 48C12 / 52C13 ] * [ 3C1 * 36C12 / 39C13 ] * [ 2C1 * 24C12 / 26C13 ]

     P ( E_1*E_2*E_3*E_4 ) = [ 4!*48! / (12!)^4 ] / [ 52! / (13!)^4 ]

     P ( E_1*E_2*E_3*E_4 ) = [ 4!*13^4 / (52*51*50*49) ]

    P ( E_1*E_2*E_3*E_4 ) = 0.1055

.

step-by-step explanation:

step-by-step explanation:

12: 00 p.m. to 7: 00 p.m. is a total of 7 hours.

4: 00 p.m. to 7: 00 p.m. is a total of 3 hours.

this means that from 4: 00 p.m to 7: 00 p.m. this represents 3/7 of what was lost from 12: 00 p.m. to 7: 00 p.m.

after knowing this, after doing 11,600 - 6,800 = 4,800

4,800 represents 3/7 of what was lost for 7 hours.

we then divide 4,800 by 3 which gets 1,600.

because we know that 1,600 represents 1/7 of how much water was lost, we just have to multiply by 7, which gets 11,200.   after adding 11,200 to how much water was left at 7: 00 p.m. which was 6,800. 11,200 + 6,800 = 18,000

this means that the pool originally had 18,000 gallons in the pool.

you can also check your work by doing 18,000 - 4(1600), which becomes           1800 - 6400 which gets you 11,600.