What is the Probability that the Lowest Card out of 4 Cards is X?

Question

If I have four different value cards, what is the probability that the lowest card (ace, lowest -> king, highest) is some value X?

Here is what I have so far:

I know that the lowest value card cannot be a king, queen, or jack, as there must exist a value that is 3 above. And I can guess that the probability of an ace will be the highest and a 10 will be the lowest. Can someone please give me some hints on how I should approach this?

Answer 1

HINT: Since you’re looking at sets of $4$ cards of different denominations, you might as well just look at sets of denominations: in effect you’re choosing $4$ cards from one suit and asking for the probability that the lowest denomination chosen is $x$. There are $\binom{13}4$ possible sets of $4$ denominations. How many have an $x$ as lowest denomination? You must choose the other $3$ from how big a set of values?

Answer 2

In this example we are going to calculate the probability of the lowest card being 3.

There are 13x12x11x10=17160 combinations of cards. There is 1 combination to get a 3 on the first card, 10 combinations to get higher than 3 on the next card, 9 combinations on the 3rd, and 8 on the 4th. The total number of combinations is 1x10x9x8. Because this can be started anywhere on the 4 cards, you multiply the result by 4, so the probability of getting any of these combinations is 10x9x8x4/17160=24/143.

We can see that in terms of $x$, which in this example is 3, the formula is $4(13-x)(12-x)(11-x)/17160=8x/143$.

Note: It could be (7+$x$).../17160 but it wouldn't work for other numbers.