# Math Help - probablility of gambling

1. ## probablility of gambling

A gambling machine shows one out of five colours, red , orange , yeloow, green and blue. When the machine is working properly, every colour has an equal chance of appearing the colour. And the colour shown at one instant is independent of the colour shown earlier. Calculate the probablity that the machine shows different colours for 5 consecutive times.

the ans is 24/625.

but my working is 0.2^5 = 1/3125

why cant i do in this way? by the way? what is the porper working of getting the ans?

2. ## Re: probablility of gambling

could you clarify this a bit

Do you just need adjacent colors to be different or do you need to see all 5 colors in your set of 5 spins?

Never mind I see you need all 5 colors.

Any permutation of all 5 colors is allowed. There are $5!$ of these.

Each individual spin has a probability of $\dfrac 1 {3125}$

$Pr[$all 5 colors in 5 spins$]=\dfrac {5!}{3125}=\dfrac{120}{3125}=\dfrac {24}{625}$

3. ## Re: probablility of gambling

0.2^5 = 1/(5^5)....5^5 is the denominator for the ratio that you want calculate, representing all the possible outcomes on the next 5 spins. 5! would be your numerator, representing the number of possible ways to get 5 different colors on the next 5 spins.

So in more detail, calculate 1) the number of possible ways you can get 5 different colors on the next 5 spins, divided by 2) the total number of possible outcomes on the next five spins (including all different colors, all the same colors, and every other possible outcome).

To calculate 1) the # of outcomes that produce 5 different colors on the next 5 spins, we can have any of the 5 colors on the 1st spin, any of the 4 remaining colors on the 2nd spin, any of the 3 remaining colors on the 3rd spin, any of the 2 remaining colors on the 4th spin, and then finally on the last spin there is only 1 possible color left that will not be the same as those on the previous 4 spins. So we have 5*4*3*2*1 = 5! as the numerator for our ratio.

To calculate 2) the total number of possible outcomes on the next 5 spins, we can have any of the 5 colors on each of the 5 spins, so the total number of outcomes on 5 spins is 5*5*5*5*5 = 5^5.

So the final answer is 5!/(5^5) = 120/(3125) = 24/625.

Another way to think about it which amounts to the same thing is as follows:

On the first spin, there is a 100% chance that you will not get a repeat of a color from a prior spin.
on 2nd spin, there is a 4/5 chance you do not get the same color that came up on the first spin.
On the 3rd spin, there is a 3/5 chance that you do not get either of the colors that came up on the 1st 2 spins.
On 4th spin, there is a 2/5 chance that you do not get any of the colors that came up on the 1st 3 spins.
On 5th spin, there is a 1/5 chance that you do not get any of the colors that came up on the last 4 spins.

So the probability of getting all different colors on the next 5 spins is 5/5 * 4/5 * 3/5 * 2/5 *1/5 = 5!/(5^5).

4. ## Re: probablility of gambling

you've made everything clear for me ! thank you very much !