5

I have 4 cards with the following numbers on them: 1, 2, 3, 4.

My opponent has 4 cards with the following numbers on them: 2, 3, 4, 6.

We both randomly draw a card from our decks, if my number is equal to or higher than my opponent's I will escape successfully. What is my probability of escaping? Please show how you arrived at the proper answer. Thanks!

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The decks here are small enough that we can explicitly list out all the possibilities:

your draw, opponent's draw
1, 2
1, 3
1, 4
1, 6
2, 2 *
2, 3
2, 4
2, 6
3, 2 *
3, 3 *
3, 4
3, 6
4, 2 *
4, 3 *
4, 4 *
4, 6

I've put an * next to all combinations which result in you escaping. As you can see there are 6 of those out of 16 total. Since each pair of draws in the list is equally likely, your probability of escaping is just 6/16 (which is 3/8 or 37.5%).

If you're trying to do this kind of calculation in the middle of a game (i.e. quickly and in your head), you may want to solve it slightly differently. First of all, we can tell that there will be 16 pairs in the list without actually listing them out, just by multiplying the deck sizes (4*4). After that you only have to count the ways to escape, which you can do by seeing how many cards each of yours beats: your 1 doesn't beat any (0), your 2 wins against their 2 (1), and so on for your 3 (2), and your 4 (3), for 0+1+2+3=6 ways of escaping.

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