Respuesta :
Answer:
An estimate of the probability that Jorge will make a profit is (5; 0.496)
Step-by-step explanation:
The total cost = 170x1 = $170
The payoff is $35 per $1 bet
The number of wins needed to make a profit = 170/35 = 4.86 \approx 5
Probability of winning, P(win), p = 1/38
n = 170
P(Jorge will make a profit) = P(at least 5 wins)
mean = np = 4.47
standard deviation = \sqrt{npq} = 2.09
P(X \geq 5) = 1 - P(X < 5)
P(X < A) = 1 - P(Z < (A - mean)/standard deviation)
After the application of continuity correction,
P(X \geq 5) = 1 - P(Z < (4.5 - 4.47)/2.09)
= 1 - P(Z < 0.01)
= 1 - 0.5040
P(X \geq 5 = 0.496
An estimate of the probability that Jorge will make a profit is (5; 0.496)
Answer:
probability that Jorge makes a profit is = 0.46412
Step-by-step explanation:
Solution:-
- The number of bets made on number "3", N = 170
- He bets on each number "3", k = $1
- The winning pay-off odds : $ ( 35 : 1 )
- The probability of getting number "3" on a spin, p = 1/38
- The total amount paid (C) for n = 170 bets on number "3" are:
C = N*k
C = (170)*($1)
C = $170
- The probability of getting a number "3" on a spin is independent for each trial.
Denote:
- The amount received per win = $ 35
- The number of wins = r
- So the minimum "N" number of wins must be enough to match loss.
Amount Win = Amount Loss
r*$35 = C
r*$36 = C
r = $170 / 36
r = 4.7222 ≈ 5 wins
- So the minimum amount of wins required by r = 10 to make a profit.
- Let a random variable "X" denote the number of times Jorge spins to get number "3" - Number of wins. The probability to get a number "3" on each spin is independent for each trial. Therefore X follows Binomial distribution.
- So, X ~ B ( N , p )
X ~ B ( 170 , 1/38 )
1 - p = 37 / 38
- So we need to determine that Jorge get number "3" at-least r = 5 times. Where the probability mass function for binomial distribution is given below:
[tex]P ( X = r ) = ^NC_r * (p)^r * ( 1 - p )^(^N^ -^ r^ )[/tex]
So,
[tex]P ( X \geq 5 ) = 1 - P ( X \leq 4) = 1 - [ P ( X = 0 ) + P ( X = 1 ) + P ( X = 2 ) + P ( X = 3 ) + P ( X = 4 )]\\\\1 - [ (37/38)^1^7^0 + 170*(1/38)*(37/38)^1^6^9 + 170C2*(1/38)^2*(37/38)^1^6^8 + \\\\170C3*(1/38)^3*(37/38)^1^6^7 + 170C4*(1/38)^4*(37/38)^1^6^6 ]\\\\1 - [ 0.01074 + 0.04935 + 0.11271 + 0.17059 + 0.19249]\\\\= 1 - 0.53588\\\\= 0.46412[/tex]
- So the probability that Jorge makes a profit is = 0.46412
Note:- The normal approximation to Binomial distribution may be a less cumbersome choice; however, care must be taken to verify the conditions for normal approximation i.e
N*p ≥ 10
With the given data, N = 170 , p = 1/38:
N*p = 170/38 = 4.4737 ≤ 10
Hence, the normal approximation is an invalid choice for the data given.
