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Best of Alan Krigman
Should you play the multi-line slots – and if so, how?16 January 2012
Machines with multiple horizontal paylines are nothing new. The classical models have three such lines. You can bet on the center, the top two, or all three. These win or lose more or less independently of one another, except for constraints like symbols in fixed positions on physical reels or numbers of stops allocated to the various images on the columns (math mavens call ‘em virtual reels). As with single-line slots, solid citizens can usually determine the results of a spin at a glance; the bells and whistles again being nice although not essential.
The current breed of multi-line slots is more elaborate. Machines commonly have five columns and three to five rows. Functionally, these devices are distinguished from their less complex predecessors by larger numbers of lines – 5, 10, 20, or more – including zig-zags as well as horizontals, passing through sets of elements in the array. Players wager on whatever number of lines as they wish. With moolah on more than a few, results of spins would be incomprehensible were it not that winning lines flash in color, legends such as “WIN WIN WIN” blast out from the screen, and the tally on the register indexes to show proceeds pumping up bankrolls.
If you have your hard-earned dough on enough lines, at least some of those bets are apt to bring home the bacon and the machine will announce you’ve won. However, while distributing a given amount across more lines raises the frequency of hits, it lowers the average amount you get back on each one. This, because returns are the net of wins on some lines minus losses on others. More regularly than you might guess, you may actually lose despite being declared a winner. For instance, you could bet $1 on each of three lines. If you get back 2-for-1 on one line while the other two come a cropper, you’ll be down $1, having started with $3 and finished with $2.
To see how this works, picture a hypothetical simplified game set up as a nine-element tic-tac-toe matrix with three rows and three columns. Say that each of the nine positions will be occupied by X or O, that the chances are 49 percent of X and the complementary 51 percent of O, and that you get back 8-for-1, a 7-to-1 payoff plus your original bet, for a line of three X’s and lose otherwise. Compare $5 on one row versus $1 each on the three rows and two diagonals.
The probability of winning on any line, irrespective of what occurs on the four others, is 0.49 multiplied by itself three times, or 11.76 percent. The likelihood of losing is the complementary 88.24 percent. If you bet on only one line, these are your prospects for joy and sorrow per spin. With $5 on one line to win $35 – collecting $40 – the house edge is (0.1176 x $35 - 0.8824 x $5)/$5, or -5.89 percent. Make believe you play this for 10,000 spins. Statistically, you expect to win $35 1,176 times and lose $5 8,824 times for a net of 1,176 x $35 - 8,824 x $5 = -$2,960.
Instead, pretend you bet $1 on each of the five lines. Your chances and the corresponding amounts are as follows.
• win all five: 0.002 percent, $35 - $0 = +$35
• win any four: 0.085 percent, $28 - $1 = +$27
• win any three:1.268 percent, $21 - $2 = +$19
• win any two: 9.508 percent, $14 - $3 = +$11
• win any one: 35.665 percent, $7 - $4 = +$3
• win none: 53.482 percent $0 - $5 = -$5
Betting on all five lines will have you smiling on an average of 46.518 percent of your spins and frowning on 53.482 percent. But, rather than $35 profit for every win at $5 on one line, the yield on fruitful spins at $1 per line will average only $5.12; 35.665 percent will get you $3 and a mere 0.002 percent will earn you $35. To get the house’s edge, multiply each indicated amount by the associated probability, add the products, then divide by the $5 total bet. It’s 5.89 percent, the same as for $5 on one line. Similarly, you can calculate how the laws of probability predict you’ll do in 10,000 spins. The net will be -$2,960, the identical loss as previously.
Since edge is constant, long-term performance doesn’t vary with the number of lines played at a given total bet per spin. In the short run, a higher rate of lower hits for a specified amount at risk reduces volatility and leads to more frequent but less profitable sessions. You may not like this if you’re bound and determined to invest resources you can readily tap on a shot at a pot of gold you doubt you can get any other way. But a modest win that doesn’t satisfy you while you’re at the casino may seem great on the way home. As that rhyming rascal, Sumner A Ingmark, wrote:
The most successful have the knack to quit when slightly in the black.
Best of Alan Krigman