Out if 56 coins, 18 are rare. If you select 10 of the coins, what us the probability that all of them are rare ?
P(picking rare coin) = 18/56 = 0.3214 q= 1-p = 1-0.3214= 0.6786 , number of trials = 10
a) All rare coins:
ncr= 10c10= n!/(r!*(n-r) = 10!/(10-10)!(10)!= ( 10!)/(0)!*(10)!= (10)!/1*(10)! = 1
P(all rare coins) = ncr*(p^r)*(q)^r = 1*(0.3214^10)*(0.6786)^10
= 1*(0.000008534)*(0.020708)=0.00000017672
disagree with that solution.
prob(rare) = 18/56 = 9/28
Prob( 10 of 10 are rare) = (9/28)^10 = .000011771
what Brandon meant was:
C(10,10) (9/28)^10 (19.28)^0
which is 1*(9/28)^10 * 1 = my answer
P(picking rare coin) = 18/56 = 0.321428571 q= 1-p = 1-0.3214= 0.6786 , number of trials = 10
a) All of them (10 trials) are rare:
ncr= 10c10= n!/(r!*(n-r) = 10!/(10-10)!(10)!= ( 10!)/(0)!*(10)!= (10)!/1*(10)! = 1
P(10 trials are rare) = ncr*(p^r)*(q)^n-r
= 1*(0.321428571)^10*(0.678571429)^10-10
= 1*(0. 321428571)^10*(0.678571429)^0
= 1*(0.0001177185)*(1) = 0.0001177185
Probability that all of them are rare = 0.0001177185
I guess this is the right answer!
The probability that all 10 selected coins are rare is approximately 0.00000017672.
To calculate the probability that all 10 coins selected are rare, we first need to calculate the probability of selecting a rare coin.
Given that there are 56 coins in total, and 18 of them are rare, the probability of selecting a rare coin is:
P(picking a rare coin) = Number of rare coins / Total number of coins
= 18 / 56
≈ 0.3214
Next, we need to calculate the probability of not selecting a rare coin (also known as selecting a non-rare coin). This can be done by subtracting the probability of selecting a rare coin from 1:
P(not picking a rare coin) = 1 - P(picking a rare coin)
= 1 - 0.3214
≈ 0.6786
Now, we can calculate the probability of selecting all 10 coins as rare using the binomial probability formula. The formula is:
P(X = r) = nCr * (p^r) * (q^(n-r))
where:
- n is the number of trials
- r is the number of successful outcomes (in this case, all 10 coins being rare)
- nCr is the combination, which can be calculated using the formula n! / (r! * (n-r)!)
- p is the probability of a successful outcome (picking a rare coin)
- q is the probability of a failure (not picking a rare coin)
Since we want all 10 coins to be rare, n = 10 and r = 10. Plugging in the values, we get:
P(all 10 coins are rare) = 10C10 * (0.3214^10) * (0.6786^0)
= 1 * (0.3214^10) * (0.6786^10)
≈ 0.00000017672
Therefore, the probability that all 10 coins selected are rare is approximately 0.00000017672.