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# Elementary?

## Question

2 items from a group of 4 can be selected in 10 distinct ways if repetitions are allowed. e.g. {(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}

Are each of these selections equally likely?

Why or why not?

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no.

There are 'technically' 16 possible outcomes some are just equivalent.

1,1 2,2 3,3 4,4 all have a 1/16 chance of occuring while the rest have a 1/8 chance.

1,1 2,2 3,3 4,4

1,2 2,1 2,3 3,2 3,4 4,3

1,3 3,1 2,4 4,2

1,4 4,1

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The question is [i think purposely] misleading

This is the combinations vs permutations questions with a little slight of
hand.

If we restate the conditions, I think it would be helpful.

The problem says "if repetitions are allowed". So the process must be
something like.

• Reach into a jar and select 1 of 4 marbles [numbered 1 thru 4]
• record the number of the first marble selected

• return the marble to the jar

• Reach into a jar and select 1 of 4 marbles [numbered 1 thru 4]
• record the number of the first marble selected

If this is fair. the distribution for each selection will be [.25, .25, .25,
.25]

When the results of the second draw are included there will be 16 possible
outcomes [permutations] and each will have a probability of 1/16.

• From a permutations perspective outcomes (12) and (21) are different and
each has a likelihood of 1/16.

• From a combinations perspective outcomes (12) and (21) are the same and this
combination has a likelihood of 1/8

The question does not specifically state which is being looked for. BUT since
it does not list (12) and (21) as outcomes, this implies that combinations is

For combinations:

NO, the outcomes are not equally likely

{(1,1),(2,2),(3,3),(4,4)} (4 elements) each occur with P = 1/16

{(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)} (6 elements) each occur with P =
1/8

For permutations:

YES. the outcomes are equally likely

{(1,1),(1,2),(1,3),(1,4),(style="color: rgb(255, 0, 0);">2,1),(2,2),(2,3),(2,4),(3,1),
(3,2),(3,3),(3,4),(4,1),(style="color: rgb(255, 0, 0);">4,2),(style="color: rgb(255, 0, 0);">4,3)(4,4)} (16 elements) each have
P=1/16

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