in the card game six in a row six cards are dealt in a row points are given for the number of consecutive cart what is the probability that the six cards are
a) consecutive and in order?
b) consecutive in any order?
To calculate the probabilities in the card game "Six in a Row," we need to first understand the total number of possible outcomes and the favorable outcomes for each case.
a) Probability of six cards being consecutive and in order:
In this case, there is only one favorable outcome because the cards need to be in a specific order. For example, if we consider the cards to be numbered from 1 to 6, the favorable outcome would be: 1, 2, 3, 4, 5, 6.
To calculate the total number of possible outcomes, we need to consider that any card can be in any position. Therefore, there are 6! (6 factorial) ways to arrange the six cards.
The formula for probability is: P(A) = favorable outcomes / total outcomes. So the probability (P) of the six cards being consecutive and in order would be:
P = 1 favorable outcome / (6! total outcomes)
P = 1 / (6!)
b) Probability of six cards being consecutive in any order:
In this case, the favorable outcomes include all possible combinations of six consecutive cards, regardless of their order. To determine the number of favorable outcomes, we can use the concept of combinations.
There are 6 cards, and any set of 6 consecutive cards can be a favorable outcome. For each card as the starting point, there is only one unique combination of six consecutive cards.
So, the number of favorable outcomes is 6.
Again, the total number of possible outcomes is 6!, or 6 factorial.
Using the probability formula, the probability of the six cards being consecutive in any order would be:
P = 6 favorable outcomes / (6! total outcomes)
P = 6 / (6!)
Note: Remember that 6! (6 factorial) means multiplying all positive integers from 1 to 6 together. So 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720.
By calculating the probabilities based on the formulas and concepts explained above, you can determine the probabilities for both cases in the card game "Six in a Row."