Answer:
Probability of selecting none of the correct six integers:
a) 0.350
b) 0.427
c) 0.489
d) 0.540
Step-by-step explanation:
a) 40
Given:
Number of integers in a lottery 6
Order in which these integers are selected does not matter
To find:
Probability of selecting none of the correct six integers
Solution:
When the order of selection does not matter then we use Combinations.
Given integers = 40
Number of ways to choose 6 from 40.
Let A be the sample space of choosing digits 6 from 40.
Then using Combinations:
(n,k) = n! / r! (n-r)!
n = 40
r = 6
40C6
=(40,6) = 40! / 6! ( 40 - 6)!
= 40! / 6!34!
= 40*39*38*37*36*35*34! / 6!34!
= 2763633600 / 720
= 3838380
Let E be the event of selecting none of the correct six integers.
So using combinations we can find the total number of ways of selecting none of 6 integers from 40
n = 40 - 6 = 34
r = 6
34C6
=(34,6) = 34! / 6! ( 34 - 6)!
= 34! / 6! 28!
= 34 * 33 * 32 * 31 * 30 * 29 * 28! / 6! 28!
=968330880 / 720
= 1344904
Probability of selecting none of the correct six integers:
P(E) = E / A
= 1344904 / 3838380
= 0.350
Probability of selecting none of the correct six integers is 0.350
b) 48
Following the method used in part a)
(n,k) = n! / r! (n-r)!
n = 48
r = 6
48C6
=(48,6) = 48! / 6! ( 48 - 6)!
= 48! / 6! ( 42 )!
= 48*47*46*45*44*43*42! / 6!42!
= 8835488640 / 720
= 12271512
Let E be the event of selecting none of the correct six integers.
So using combinations we can find the total number of ways of selecting none of 6 integers from 48
n = 48 - 6 = 42
r = 6
42C6
= (42,6) = 42! / 6! ( 42 - 6)!
= 42! / 6! 36!
= 3776965920
= 5245786
P(E) = E / A
= 5245786/12271512
= 0.427
c) 56
(n,k) = n! / r! (n-r)!
n = 56
r = 6
56C6
=(56,6) = 56! / 6! ( 56- 6)!
= 56! / 6! ( 50 )!
= 56*55*54*53*52*51*50! / 6! 50!
= 23377273920/6
= 32468436
Let E be the event of selecting none of the correct six integers.
So using combinations we can find the total number of ways of selecting none of 6 integers from 56
n = 56 - 6 = 50
(50,6) = 50! / 6! ( 50- 6)!
= 50*49*48*47*46*45*44! / 44! 6!
= 11441304000 / 6
= 15890700
P(E) = E / A
= 15890700 / 32468436
= 0.489
d) 64
(n,k) = n! / r! (n-r)!
n = 64
r = 6
64C6
=(64,6) = 64! / 6! ( 64 - 6)!
= 64! / 6! ( 58 )!
= 64*63*62*61*60*59*58! / 6! 58!
= 53981544960 / 720
= 74974368
Let E be the event of selecting none of the correct six integers.
So using combinations we can find the total number of ways of selecting none of 6 integers from 64
n = 64 - 6 = 58
(58,6) = 58! / 6! ( 58- 6)!
= 58*57*56*55*54*53*52! / 52! 6!
= 29142257760/ 6
= 40475358
P(E) = E / A
= 40475358/ 74974368
= 0.540
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