Probablility of a gambling machine

In summary: So you should also discard 0.2^5 which is the probability of obtaining a particular permutation.In summary, the correct way to approach this problem is to first calculate the total number of possible sequences that can be formed using 5 different colors, which is 5! = 120. Then, we need to calculate the number of sequences that fulfill the condition of not repeating any color, which is also 5!. Thus, the probability of the machine showing different colors for 5 consecutive times is 5!/5! = 1, or 100%.
  • #1
somecelxis
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0

Homework Statement


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 can't i do in this way? by the way? what is the porper working of getting the ans?


Homework Equations





The Attempt at a Solution

 
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  • #2
Can you work out the probabability that the machine shows red, orange, yellow, green and blue in that order? Do you still think ## 0.2^5 ## can be the right answer?

If the machine is going to show different colours for 5 consecutive times, does it matter what colour it shows the first time? How many choices of coulour can it show the second time?
 
  • #3
MrAnchovy said:
Can you work out the probabability that the machine shows red, orange, yellow, green and blue in that order? Do you still think ## 0.2^5 ## can be the right answer?

If the machine is going to show different colours for 5 consecutive times, does it matter what colour it shows the first time? How many choices of coulour can it show the second time?

sorry. i still can't get what do you mean . can you explain in other words?
 
  • #4
Let the first choice be arbitrary. Any color. Then how many choices would you have for the second color if you have to avoid duplicating any color?
 
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  • #5
If the machine has shown red on the first try, what is the probability that it will show a different color (any color different from red) in the second try?
If the machine has shown red on the first try and blue on the second try, what is the probability that it will show a color different from red and blue on the third?
Take it from there.

Your expression ##0.2^5## is the probability of obtaining a particular sequence (say red, blue, yellow, orange, green), but there are other sequences that also show all five colors.
Edit: You can actually use this to solve the problem as well by computing the number of different sequences that fulfill your condition of not repeating any color.
 
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  • #6
Orodruin said:
If the machine has shown red on the first try, what is the probability that it will show a different color (any color different from red) in the second try?
If the machine has shown red on the first try and blue on the second try, what is the probability that it will show a color different from red and blue on the third?
Take it from there.

Your expression ##0.2^5## is the probability of obtaining a particular sequence (say red, blue, yellow, orange, green), but there are other sequences that also show all five colors.
Edit: You can actually use this to solve the problem as well by computing the number of different sequences that fulfill your condition of not repeating any color.

WHY NOT 1/(5x4x3x2x1)= 1/120 ...?
after a certain colour(from n choices) is chosen for the first attempt , i have n-1 choices for the subsequent attempt , where n = 5,4 ,3,2 ,1
 
  • #7
somecelxis said:
WHY NOT 1/(5x4x3x2x1)= 1/120 ...?
after a certain colour(from n choices) is chosen for the first attempt , i have n-1 choices for the subsequent attempt , where n = 5,4 ,3,2 ,1

This answer is incorrect. The number 12 = 5! is the number of permutations of 5 different things, so 1/120 is the probability of a PARTICULAR permutation. However, that is not what the question asks for. In your question you are trying to count only a subset of all possible outcomes---those that actually DO constitute a permutation of the 5 numbers. In other words, you want to discard those outcomes in which one or more of the numbers is repeated more than once, and look only at the ones that constitute permutations.
 

1. What is the probability of winning on a gambling machine?

The probability of winning on a gambling machine varies depending on the specific machine and game being played. However, most gambling machines are designed to give the house an edge, meaning the probability of winning is typically less than 50%. This ensures that the casino or establishment will make a profit in the long run.

2. How is the probability of a gambling machine determined?

The probability of a gambling machine is determined by the game's payout structure and the odds of each outcome. For example, in a slot machine, the probability of winning a jackpot may be 1 in 10,000, while the probability of winning a smaller prize may be 1 in 100. These probabilities are carefully calculated by the machine's creators to ensure the desired level of profitability for the casino.

3. Can the probability of a gambling machine be manipulated?

In most cases, the probability of a gambling machine cannot be manipulated by the player. These machines are heavily regulated and undergo regular audits to ensure fairness and accuracy. However, there have been instances of individuals attempting to cheat the system, but this is illegal and can result in severe penalties.

4. Does the probability of a gambling machine change over time?

The probability of a gambling machine does not change over time. Each spin or play is independent of the previous one, and the machine's odds remain the same. However, the longer a person plays on a gambling machine, the higher the chances of losing money due to the house's edge.

5. Is there a way to increase the probability of winning on a gambling machine?

Unfortunately, there is no guaranteed way to increase the probability of winning on a gambling machine. These machines operate using random number generators, so each outcome is entirely unpredictable. However, some strategies, such as setting a budget and sticking to it, can help players manage their money and potentially increase their chances of walking away with a profit.

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