Risk & Reward

A Five-Section Wheel

A five-section wheel pays $1 on four sections and $5 on the fifth, and a spin costs $1.50. Should you play if you can spin as often as you like, and should you play if you get exactly one spin?

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A fair wheel has five equal sections. Four of them pay $1 and the fifth pays $5. Each spin costs $1.50. Part one, if you may play as often as you want, should you play? Part two, if you may play exactly once, should you play?

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#Expected payout of a spin

Each section is equally likely, so the average payout is

E[payout]=4⋅1+1⋅55=95=1.80.(1)\E[\text{payout}] = \frac{4 \cdot 1 + 1 \cdot 5}{5} = \frac{9}{5} = 1.80. \tag{1}

A spin returns $1.80 on average against its $1.50 price, an edge of $0.30 per spin.

$5$1$1$1$1
Four sections pay a dollar and one pays five, averaging one dollar eighty per spin. That sits above the dollar fifty ticket, a thirty cent edge that pays whether you spin once or a thousand times.

#Part one, playing repeatedly

Every spin carries the same positive edge, so the law of large numbers drives your average profit toward $0.30 a spin. Over many spins you come out ahead almost surely, so play as often as you can.

#Part two, playing once

A single spin has the same expectation. Its profit is a $0.50 loss with probability 45\tfrac45 and a $3.50 gain with probability 15\tfrac15,

E[profit]=45(−0.50)+15(3.50)=−0.40+0.70=0.30>0.(2)\E[\text{profit}] = \tfrac45(-0.50) + \tfrac15(3.50) = -0.40 + 0.70 = 0.30 > 0. \tag{2}

The wager is favourable even taken once, so a player who maximises expected value still spins.

#Result

Yes in both cases. The wheel pays $1.80 on average for a $1.50 ticket, a positive edge that rewards a single spin and compounds across many.