"What Are The Odds?" Topic

Hello

Query from a maths dunce here.

I'm OK with the odds of an event happening in a single roll (eg to roll a particular number on a d6 is 1/6). However, I get confused by the possibilities when modelling a number of events.

For example: using a system using an ordinary set of playing cards the possibilities of drawing, say, a six or less (from the full deck) is 6/13 or roughly 46% (ace = 1).

Using this system, say, in a WW2 game where Tank A fires at Tank B and needs to firstly hit, then penetrate then check for effect, how do I calculate the possibility of getting a kill from the outset?

EG To hit needs to draw a 7 or less:
To penetrate then needs to draw a 4 or less
To kill requires a 5 if penetrated.

Am I right in thinking the odds are:

7/13 x 4/13 x 5/13 = about 6%? (not much of a shot!)

(I am aware that the odds will change minutely when drawing cards in sequence as the deck decreases but I am not concerned to that extent)

I just want to know if my methodology is correct.

Thanks

Nelclaret

Your methodology is correct, but don't discount the skewing effect on the odds if you aren't replacing the cards. It is

28/52 x 16/51 x 20/50 = 0.066 (near enough).

Closer to 7% than 6% – the difference being non replacement of the cards.

The overall probability of success is the probability of each event multiplied by the preceding one.

You are correct.

Martin: skewing will work both ways. You need a low card to be successful; in order to progress you may be removing potential successes from the deck. So you progressively get a smaller deck, but with fewer successes.

Using your example, if the first two cards drawn are 5 or less, the oddds are
28/52 x 15/51 x 18/50 = 0.057%

In this case as the succeeding chances are lower, there will be some successes that don't lower your chance – I'd say 6% is close enough.

Advocate: you gave the answer as a decimal and then tacked a % onto the end.

It's actually 5.7%.

Written as a probability, it's 0.057 (no "%" sign).

Nelclaret: as a rule, game designers don't understand cumulative probability, leading to some nasty, unwanted effects. An example is when unit morale declines from damage, and you have to make a morale check on every point of decline. If your starting morale is 11 and you check for that or lower on 2d6, how many morale hits do you have to take before your cumulative chance of failure is 50% or more? Not nearly as many as you'd think …

Gentlemen

Thank you for your replies. I'm glad I worked out the method correctly.

I would like to pose a 'supplemental', if I may (and its a doozy!)

I have a set of modern jet combat rules. This uses a grid of squares and a pack of playing cards.

When an aircraft fires a missile at a target the missile moves across the grid to try and reach the target square.

The movement of the misile is handled by drawing cards from the deck, and the missile moves the distance in squares equal to the difference in value between the cards drawn.

Missiles will draw a different number of cards ('Turns of Flight')depending on whether it is short, medium or long-ranged. The sequence starts with the drawing of a reference card, and then the drawing of a number of cards equal to the missile's 'Turns of Flight'

(In this system an Ace = 1, Jack = 11, Queen = 12, King = 13 and all other cards are at face value).

An example should clarify:

Imagine a Missile is fired at a target. The missile has 3 Turns of Flight. A reference card is drawn. It is a 6.

The first card of the missile's turns of flight is drawn and is a 3. The missile moves (6-3 =) 3 squares.

The next card is a Queen (value 12). This gives a further (12-3 =) 9 squares of movement

The final card is a 9. This gives (12-9 =) 3 squares of movement.

In total the missile has moved 15 squares. If the target is within 15 squares then there is a possible hit.

I hope this is reasonably clear.

My question: is it possible (without using NASA's bank of computers!) to determine the average/likely/mean distance the missile will move once the reference card is drawn?

I have thought about this for a while and my brain hurts!

I realise that there may be some distortion if the cards are not replaced in the pack after each drawing.

An added complication (!) : If two consecutive cards of the same value are drawn the missile is deemed to have malfunctioned and automatically misses.

I doubt this is soluable in easy terms but I thought I would ask.

If you've got this far, thanks for reading!!

NL

My question: is it possible (without using NASA's bank of computers!) to determine the average/likely/mean distance the missile will move once the reference card is drawn?

Of course. This is a fairly trivial, though long-winded thing to solve exactly.

On the whole such things are best set up by Monte Carlo methods. That is, a lot of trials. That allows you to observe the spread and alter the parameters very quickly to try variants.

I would do it in Excel myself, using a Macro to draw cards and compute results. Run it a couple of thousand times until the moving average stabilised.

It wouldn't take very long to program if you know what you are doing. And if you don't then you would be learning some useful skills. <grin>

Thinking about it, there are only a touch under a million alternative ways of drawing 4 cards. You could get most programs to do that fairly quickly the long way.

And I'm puzzling over why the missiles speed changes so radically. And what is the minimum air speed to stay aloft? Most of the air-to-air missiles would have glide paths like rocks without high initial speeds.

Resolution mechanics in games shouldn't be looked at too closely. (Repeat until calm…)

[An added complication (!) : If two consecutive cards of the same value are drawn the missile is deemed to have malfunctioned and automatically misses]

This rule will make missiles malfunction 55.7% of the time.