Double Trouble

I heard an ad for a sports betting app offering the “chance” to recover your bet if the player you picked to score his team’s first touchdown didn’t do it, but did score the second. The idea behind this presumably being that someone will feel more comfortable placing a first bet if they think there’s a chance they can recoup their losses later.

This hinges on some pretty basic misunderstandings of probability. Let’s start with expected winnings. Ashton Jenty had the highest share of his team’s touchdowns with 42%, though the typical player ranged from 0% to 14%. Let’s say we felt confident about Janty and bet $100 on him. We’d expect that, if we did that 100 times, we’d win 42% of those bets and lose 58%. $100 * 42% is $42 won, and $100 * $58 is $58 lost. $42 - $58 is -$16, so we’d expect to lose money long term, even with the most likely player.

Ah, but the ad says in those 58% of scenarios, we could recup if Jenty scores the second touchdown. What would that do to our expected outcomes? Well, we know it would happen 58% of the time, and we expect Janty to score 42% of his teams touchdowns. So 58% * 42% = 24.36%*. Multiply that by $100, and we get $24.36 for our second round. Add that to -$16, and we get $8.36.

*This is quasi-conditional probability. Not exactly Bayesian, but not exactly not Bayesian, either.

Before we go crazy here, let’s recall that we were looking at the best case scenario. Let’s instead break it down for anything under 50%.

library(hrbrthemes)
library(janitor)
library(gt)
library(gtExtras)
library(tidyverse)

odds <- rep(seq(0.00, 0.5, by = 0.05)) %>%
  as_tibble() %>%
  rename(odds = value)

odds_two <- odds %>%
  mutate(bet_value = 100.00,
         exp_winnings = (odds * bet_value),
         exp_losings = -((1 - odds) * bet_value),
         exp_total = exp_winnings + exp_losings,
         exp_odds_two = odds * (1-odds),
         exp_win_two = exp_odds_two * bet_value,
         exp_total_two = abs(exp_win_two) - abs(exp_total)) 

odds_two %>%
  select(-bet_value) %>%
  gt() %>%
  gt_theme_espn() %>%
  fmt_currency(columns = c('exp_winnings', 'exp_losings', 'exp_total', 'exp_win_two', 'exp_total_two')) %>%
  fmt_percent(c('odds', 'exp_odds_two'), decimals = 1) %>%
  tab_spanner('First TD', exp_winnings:exp_total) %>%
  tab_spanner('Second TD', exp_odds_two:exp_total_two) %>%
  cols_label(exp_winnings = 'Expected Win',
             exp_losings = 'Expected Loss',
             exp_total = 'Expected Total',
             exp_odds_two = 'Odds',
             exp_win_two = 'Expected $',
             exp_total_two = 'New Total Expected $')

odds	First TD			Second TD
	Expected Win	Expected Loss	Expected Total	Odds	Expected $	New Total Expected $
0.0%	$0.00	−$100.00	−$100.00	0.0%	$0.00	−$100.00
5.0%	$5.00	−$95.00	−$90.00	4.8%	$4.75	−$85.25
10.0%	$10.00	−$90.00	−$80.00	9.0%	$9.00	−$71.00
15.0%	$15.00	−$85.00	−$70.00	12.8%	$12.75	−$57.25
20.0%	$20.00	−$80.00	−$60.00	16.0%	$16.00	−$44.00
25.0%	$25.00	−$75.00	−$50.00	18.8%	$18.75	−$31.25
30.0%	$30.00	−$70.00	−$40.00	21.0%	$21.00	−$19.00
35.0%	$35.00	−$65.00	−$30.00	22.7%	$22.75	−$7.25
40.0%	$40.00	−$60.00	−$20.00	24.0%	$24.00	$4.00
45.0%	$45.00	−$55.00	−$10.00	24.8%	$24.75	$14.75
50.0%	$50.00	−$50.00	$0.00	25.0%	$25.00	$25.00

Note that, while our odds may improve slightly, we still expect to lose most of the time. Even a 20% share player only has a 16% chance of being the second to score a TD, so all you’re really doing in this case is bundling losing bets.

Even the Janty example is overoptimistic, since we just looked at total share of touchdowns, not first in game. This is, of course, subject to randomness, but I’m willing to bet that a check of the Raider’s logs will show Janty scoring more than the first TD.

library(ggtext)

odds_g <- rep(seq(0.00, 0.5, by = 0.01)) %>%
  as_tibble() %>%
  rename(odds = value)

odds_g_two <- odds_g %>%
  mutate(bet_value = 100.00,
         exp_winnings = (odds * bet_value),
         exp_losings = -((1 - odds) * bet_value),
         exp_total = exp_winnings + exp_losings,
         exp_odds_two = odds * (1-odds),
         exp_win_two = exp_odds_two * bet_value,
         exp_total_two = abs(exp_win_two) - abs(exp_total)) 

odds_g_two %>%
  ggplot(aes(x = odds, y = exp_total))+
  geom_line(color = 'blue')+
  geom_line(aes(y = exp_total_two), color = 'darkgrey')+
  theme_ipsum()+
  theme(text=element_text(size = 16,  family="Oswald"),
        panel.grid.minor = element_blank(),
        plot.subtitle = element_markdown(),
        plot.title.position = "plot")+
  scale_y_continuous(labels = scales::label_number(scale_cut = scales::cut_short_scale()))+
  ylab('')+
  xlab('')+
  labs(title = 'Expected Winnings on TD Bets',
       subtitle = "Comparing <span style='color:blue'>TD One</span> and <span style='color:darkgrey'>TD Two</span>")

Note that, even though the bets get less risky, odds are still in the house’s favor. Sports Betting sites are always going to win in the long run, it wouldn’t be much of a business model for them if they didn’t. This ad is a ploy to make something risky seem safe, and counts on the often confusing nature of probabilities to get people’s interest. Take the time to understand the odds first, then put your money into a more secure savings system instead of betting it away.