MHB Dice Probability: 5 Rolls, Increasing Number

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The probability of rolling a higher number than the previous roll in five consecutive dice rolls is calculated by considering the unique increasing sequences possible with six outcomes. Since all rolls must be different, there are exactly six valid increasing sequences. The total number of possible sequences for five rolls is \(6^5\). Thus, the probability is determined by the ratio of the six increasing sequences to the total possible sequences. This results in a clear understanding of the likelihood of achieving such an outcome in five rolls.
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A Dice is Rolled $5$ times. The Probability of Getting a higher number then the previous number each time is
 
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It's easy to explicitly enumerate all increasing sequences of 5 numbers from {1, ..., 6}.
 
jacks said:
A Dice is Rolled $5$ times. The Probability of Getting a higher number then the previous number each time is

Because the sequences are increasing all the die rolls are different, as there are only 6 possible outcomes all but one of the outcomes must be present in such a sequence and there is only one order that these can be in, so there are exactly \(6\) increasing sequences of 5 die roll outcomes from a total of \(6^5\) possible sequences of 5 die roll outcomes...

CB
 

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