π₯ π
°π
±π
°π
±π
°π
±
1/6 β«β«βͺβͺβͺβͺ
1/6 βͺβ«β«βͺβͺβͺ
1/6 βͺβͺβ«β«βͺβͺ
1/6 βͺβͺβͺβ«β«βͺ
1/6 βͺβͺβͺβͺβ«β«
1/6 β«βͺβͺβͺβͺβ«
π
°π«
β»οΈ π₯β‘οΈπ
°β°οΈ
β»οΈ ββ‘οΈπ
π
°π«π
±
β»οΈ π₯β‘οΈπ
°β°οΈ
β»οΈ ββ‘οΈπ
°πͺπ
±β‘οΈπ
±β°οΈ
βββββ (π
°)
1/1 βͺβͺβͺβͺβ«β«
π
°π«
β»οΈ 1/1 π₯β‘οΈπ
°β°οΈ
π
±π«π
°
β»οΈ 1/1 π₯β‘οΈπ
±β°οΈ
βοΈπ
±π«π
°β‘οΈπ―π
±β°οΈ
ββββ (π
±)
1/2 βͺβͺβͺβ«β«βͺ
1/2 βͺβͺβͺβͺβ«β«
π
±π«
β»οΈ 1/2 π₯β‘οΈπ
±β°οΈ
β»οΈ 1/2 ββ‘οΈπ―π
±β°οΈ
π
°π«π
±
β»οΈ 1/2 π₯β‘οΈπ
°β°οΈ
β»οΈ 1/2 ββ‘οΈπ
°πͺπ
±β‘οΈπ
±β°οΈ
βοΈπ
°π«π
±β‘οΈ1/2π
°β°οΈβ1/2π
±β°οΈ
βββ (π
°)
1/3 βͺβͺβ«β«βͺβͺ
1/3 βͺβͺβͺβ«β«βͺ
1/3 βͺβͺβͺβͺβ«β«
π
°π«
β»οΈ 1/3 π₯β‘οΈπ
°β°οΈ
β»οΈ 2/3 ββ‘οΈ1/2π
°β°οΈβ1/2π
±β°οΈ
π
±π«π
°
β»οΈ 1/3 π₯β‘οΈπ
±β°οΈ
β»οΈ 2/3 ββ‘οΈπ
±πͺπ
°β‘οΈπ
°β°οΈ
βοΈπ
°π«/π
±π«π
°β‘οΈ2/3π
°β°οΈβ1/3π
±β°οΈ
ββ (π
±)
1/4 βͺβ«β«βͺβͺβͺ
1/4 βͺβͺβ«β«βͺβͺ
1/4 βͺβͺβͺβ«β«βͺ
1/4 βͺβͺβͺβͺβ«β«
π
±π«
β»οΈ 1/4 π₯β‘οΈπ
±β°οΈ
β»οΈ 3/4 ββ‘οΈ2/3π
°β°οΈβ1/3π
±β°οΈ
π
°π«π
±
β»οΈ 1/4 π₯β‘οΈπ
°β°οΈ
β»οΈ 3/4 ββ‘οΈπ
°πͺπ
±β‘οΈπ
±β°οΈ
βοΈπ
±π«β‘οΈ1/2π
°β°οΈβ1/2π
±β°οΈ
β (π
°)
1/6 β«β«βͺβͺβͺβͺ
1/6 βͺβ«β«βͺβͺβͺ
1/6 βͺβͺβ«β«βͺβͺ
1/6 βͺβͺβͺβ«β«βͺ
1/6 βͺβͺβͺβͺβ«β«
1/6 β«βͺβͺβͺβͺβ«
π
°π«
β»οΈ 1/3 π₯β‘οΈπ
°β°οΈ
β»οΈ 2/3 ββ‘οΈ1/2π
°β°οΈβ1/2π
±β°οΈ
π
±π«π
°
β»οΈ 1/3 π₯β‘οΈπ
±β°οΈ
β»οΈ 2/3 ββ‘οΈπ
±πͺπ
°β‘οΈπ
°β°οΈ
βοΈπ
°π«/π
±π«π
°β‘οΈ2/3π
°β°οΈβ1/3π
±β°οΈ
β2/3π
°β°οΈ
βοΈ1/3π
±β°οΈ
I'll call the player who goes first A and the one who goes second B.
On the fifth turn it's A's move. If the game reaches this point, A
knows that a round is chambered, so he will always opt to shoot B.
On the fourth turn it's B's move. At this point the odds that a round
is chambered is 1:1 even. B knows if he fires at himself he will
either die now or in the next round, so he opts to fire at A, where
his odds of survival are 1:1 even.
On the third turn it's A's move. At this point the odds that a round
is chambered is 2:1 against. It makes no difference what A does: if
he fires at himself he dies now with probability 1/3, or dies in the
next round with probability 1/2, leading to an odds of survival of 2:1
against. If he fires at B his odds are the same, 2:1 against.
On the second turn it's B's move. At this point the odds that a round
is chambered is 3:1 against. If B fires at himself he dies now with
probability 1/4, or dies in a subsequent round with probability 1/3,
leading to an odds of survival of 1:1 even. If he fires at A his odds
are 3:1 against, so he chooses to fire at himself.
On the first turn it's A's move. At this point the odds that a round
is chambered is 2:1 against. It makes no difference what A does: if
he fires at himself he dies now with probability 1/3, or dies in a
subsequent round with probability 1/2, leading to an odds of survival
of 2:1 against. If he fires at B his odds are the same, 2:1 against.
Thus B has the higher chance of survival, with odds of 2:1 on.