## When to double?

This makes certain simplifying assumptions that won't hold in practice:

• the contract will either make exactly or go off by one trick;
• the opponents will not redouble or run to another contract;
• the act of doubling won't change the number of tricks made;
• in pairs or teams scenarios, all pairs / the other team are facing the same decision.
Probability of beating contract needed for double to be profitable
Non-vulnerable Vulnerable

## When to redouble?

Probability of making contract needed for redouble to be profitable
Non-vulnerable Vulnerable

## When to bid game?

Probability of making game needed to make bidding it profitable
Non-vulnerable Vulnerable

## When to bid small slam?

Probability of making small slam needed to make bidding it profitable
Non-vulnerable Vulnerable

## When to bid grand slam?

Probability of making grand slam needed to make bidding it profitable
Non-vulnerable Vulnerable

## When to sacrifice over making game?

Required probability of going only n off vs. n+1 off
Green (n = 3) White (n = 2) Amber (n = 2) Red (n = 1)

## When to sacrifice over making small slam?

Required probability of going only n off vs. n+1 off
Green (n = 5) White (n = 4) Amber (n = 4) Red (n = 3)

As you can see, six of the cases above have 0% probability requirement. This is due to 1370 and 1430 (or 1440) lying on either side of 1400 (6 off doubled non-vulnerable or 5 off doubled vulnerable). This leads to different n values as shown in the following table:

Required probability of going only n off vs. n+1 off
Green (n = 6) White (n = 4) Amber (n = 5) Red (n = 3)

## When to sacrifice over making grand slam?

Required probability of going only n off vs. n+1 off
Green (n = 8) White (n = 6) Amber (n = 7) Red (n = 5)