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Elementary?

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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
what is being asked for.

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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