This article examines the Wallace case using "forensic mathematics" so please be warned: if you are arithmophobic please click away now! It also assumes you are familiar with the case and have read Move To Murder.
On Monday 19 January 1931, William Herbert Wallace said he left his house, as far as he could tell, at 7:15pm. He headed south, walked along Richmond Park to Breck Road, turned left and boarded a tram by Belmont Road. The prosecution at his trial insisted this was a lie, claiming he actually walked north to the call box in Lower Breck Road and boarded a tram at a stop close by after leaving the infamous message at his chess club.
Of all the evidence in the Wallace case, arguably the night of the call provides us with the largest set of quantitative data, enabling a more objective analysis. Are the circumstances surrounding the call and Wallace's arrival time at the chess club an indicator that Wallace made the infamous call, or the opposite?
In mathematics, likelihood is different to probability. The latter is a widely-understood concept and answers such questions such as What is the chance a 7 will be rolled with a single throw of two dice? Likelihood is, roughly, taking an outcome and asking what were the chances of the known result given a particular situation. For example, if we know a 7 was obtained on a single throw, we might be interested in knowing how many dice were rolled. We realise immediately that the answer must be between two and seven because it is impossible to total 7 with any other number of dice. And the greatest likelihood is that the total was obtained with two. The odds of rolling a 7 diminish with every die added.
To our question. What is the likelihood that Wallace arrived at the chess club at "about 7:45 pm" after he made the call? In particular, is it more or less likely compared to him not making the call? To answer this we will build a simple mathematical model based on the parameters of the situation. And these are as follows.
Wallace claimed he left his house for the chess club at "about 7:15 pm". For now we assume he left at precisely this time but, as I will show later, he almost certainly left a bit earlier.
We know the call was put through to the City Cafe at 7:20 pm. Annie Robertson, the Anfield Telephone Exchange supervisor, said she fixed the time by the exchange clock, which we assume was accurate.
We do not know precisely when the call ended, but obviously it can be calculated by estimating its duration. Author Jonathan Goodman assumed it was four minutes and attempted to show that Wallace could not make the chess club in time. However, in my book I revealed that it was possible for Wallace to have made the call and reach the chess club by tram even assuming the call lasted this long. Having acted out the likely sequence of the conversation, as described in my book, I believe the call lasted two to three minutes, including the time it took the waitress to fetch Beattie.
Whether Wallace made the call or not, he travelled into central Liverpool on the No. 14 tram, which was running every 8 - 9 minutes that night (and less frequently than normal due to subsidence in Dale Street). We know this from the engineer's report (see Exhibit 3 in Move to Murder). Phenomena like tram waiting times typically follow a uniform probability distribution; the chance of waiting one minute is the same as waiting nine minutes; we certainly have no reason to believe it was otherwise on this night. Therefore, we assume the wait time ranged equally from 1 to 9 minutes. The lower bound is not zero to allow for the short walk to the tram stop and boarding, which delays the tram from leaving immediately.
The engineer found it took 24 minutes to travel from 29 Wolverton Street to the City Cafe in central Liverpool, including the 533-yard walk to the Belmont Road junction where Wallace claimed he boarded the tram (Route 1). It took a little over 20 minutes to travel from the Anfield call box to the cafe, the route the police believed Wallace took (Route 2). Both journey times exclude tram waiting and boarding.
Wallace arrived at the chess club at "about 7:45 pm" on the Monday night. This was confirmed by James Caird. Because of the club rule stating that players could be penalised for turning up later than this for a tournament game, some believe Wallace must have arrived before then. To me, this sounds like a classic mistake of the armchair detective, assuming reality is similar to the clockwork world of crime fiction. A few minutes either side will have made little difference, especially as the match Wallace actually played was unscheduled for that evening and no one was waiting for him. However, for now we will assume he arrived at precisely 7.45 pm.
So, we have our initial set of parameters.
|TABLE 1 - INITIAL PARAMETERS|
|Journey Route 1||24 mins|
|Journey Route 2||20 mins|
|Tram Wait||1-9 mins|
|Call Duration||2-3 mins|
We can construct a very simple model for both routes. The only variable affecting the total journey time of Route 1 is Tram Wait, so we have nine possibilities, and only one of these allowed Wallace to arrive at precisely 7:45 pm. Route 2 is also affected by Call Duration. Only two of the 18 possibilities allowed Wallace to arrive at 7.45 pm. The two have identical likelihoods:
|TABLE 2 - INITIAL LIKELIHOODS|
|Pr (Truth: Wallace was not the caller)||0.11|
|Pr (Lie: Wallace was the caller)||0.11|
|Mean Tram Wait||2'30"|
The actual values, expressed as probabilities, are not important. We're interested in the ratio between the two, and we see it was equally likely for Wallace to arrive at the chess club by either route. We also note that Wallace had a longer wait time if he told the truth, a shorter one if he lied and made the call, but each is as likely as the other.
Caird confirmed that Wallace arrived at the chess club at "about 7.45 pm". We cannot assume he arrived precisely at this time and need to allow a margin of error. I suggest three minutes either side is reasonable, i.e. the arrival time is a seven minute range between 7.42 and 7.48 pm. I would say that if Wallace actually turned up at any of these times it is consistent with him arriving at about 7.45 pm.
There is other evidence, often overlooked, which should also be assessed to derive a range of arrival times. The first is that Beattie stated that he gave Wallace the message about half an hour after the phone call ended. We believe the call ended between 7.22 and 7.23 pm, implying the message was relayed at about 7.52 - 7.53 pm. Again, we should allow a margin of error of three minutes either side, giving a range of 7.49 - 7.56 pm.
In his first police statement, Wallace recalled that Beattie delivered the message about 10 minutes into the game, while McCartney stated it was between 5 and 10 minutes. It is reasonable to assume Beattie delivered the message about 8 minutes into the game (the mean is 8'45"). Hence, this evidence suggests the chess match began between 7.41 and 7.48 pm.
We are interested in Arrival Time, and must account for Wallace's activities at the club prior to starting the match. How quickly could he hang his coat and hat, scan the schedule on the notice board, talk to Caird, establish Chandler was a no-show, find McCartney and begin play? I think it is reasonable to assume two minutes, which suggests Wallace arrived sometime between 7.39 pm and 7.46 pm. However, we must also be consistent with the original estimates of arrival from both Caird and Wallace. Hence, we assume Arrival Time to be between 7.42 and 7.46 pm. In effect, the evidence of the message delivery shaves two minutes from the upper bound of the original range. So, we have changed one parameter from Table 1:
|TABLE 3 - CHANGED PARAMETER|
|Parameter||Value||Arrival Time||7:42 - 7:46|
This range of times is based on the statements of Wallace (twice), Caird, Beattie and McCartney. Of course, it is possible that any of these witnesses were mistaken and Wallace arrived at a different time, but I suggest we have to accept the evidence as we have it unless there is reason to believe otherwise. And with this parameter change, we find the following:
|TABLE 4 - ADJUSTED LIKELIHOODS|
|Pr (Truth: Wallace was not the caller)||0.56|
|Mean Arrival Time||7:44.00|
|Mean Tram Wait||5'|
|Pr (Lie: Wallace was the caller)||0.39|
|Mean Tram Wait||2'17"|
|Likelihood Ratio (approx)||7:5|
It is now more likely that Wallace told the truth. The difference is not massive, so we should say the timings marginally favour Wallace not making the call. So, Arrival Time affects the likelihoods, but what about Departure Time? The scenario in which Wallace made the call is unaffected because there is a fixed point on that timeline: the 7.20 pm logging of the call. Wallace could have left his house as early as, say, 6.30 pm, gone somewhere else first and still made the call at 7:20 pm. But what affect does it have on the non-call scenario?
We cannot know that Wallace left his house at precisely 7.15 pm so defining a range of times is appropriate. Again we could use three minutes either side, i.e. assume Wallace left between 7.12 and 7.18 pm, but there is a problem. Someone made the call, and if it was not Wallace then all plausible theories suggest it was Parry after he observed Wallace leaving for the chess club. But if Wallace left as late as 7.18 pm, we have to allow some time for Wallace to walk to a location which Parry had staked out and for Parry to get to the call box. This would mean Parry arrived at the kiosk at 7:19 at the earliest, and that does not fit with the chronology of the call.
During the committal hearing, operator Louisa Alfreds said the initial call was made at 7.15 pm. I wonder if the call actually came through a minute or two later , but let's assume it was made at 7.15 pm and by Parry. If the young actor waited somewhere, say, in Richmond Park  he could observe Wallace shortly after he left his house and then drive the short distance to the call box. This could all be achieved within two minutes (n.b. it took four minutes to walk from Wallace's house to the call box). So, we now have changed two parameters:
|TABLE 5 - CHANGED PARAMETERS|
|Arrival Time||7:42 - 7:46|
With the change in Departure Time, the model shows everything remains the same as in Table 4 apart from Mean Tram Wait for the scenario in which Wallace was not the caller; it increases to 7 minutes. Mathematically, the earlier Departure Time is compensated by a greater Tram Wait.
Now this might seem like smoke and mirrors, but the model is telling us something significant. The average tram wait is three times longer if Parry made the call. In fact, given our assumptions, if Wallace had a tram wait of five minutes or more, Parry was the caller. On the other hand, if Wallace waited four minutes or less, Wallace stepped into the phone box.
We've deduced a fact that would have been a big pointer in determining whether Wallace was telling the truth or not. Had he been asked, I expect Wallace would have answered truthfully; I simply don't believe he could have instantly worked out the mathematics, and probably thought the question was trivial anyhow. So, if this were a novel, our hero detective would turn to Wallace and say, "Ah, one last thing before I leave, Mr Wallace. You said you boarded a tram at Belmont Road?"
"Yes, that is so."
"How long did you wait for it?"
"Oh, it was about...."
We cannot hear his answer, but the detective nods and leaves.
I'm sorry I cannot provide the ending we all crave because we do not know the answer to that question. But is there a clue in his movements that night? Did Wallace first wait at the stop at the end of Richmond Park or Newcombe Street, but after waiting several minutes in the cold January night air decided to walk on to the next stop at Belmont Road to keep warm? But if this were the case, why didn't he mention this fact? Or did he put the phone down, anxious the call had taken longer than anticipated, hurried across the road, and was relieved to see the tram already whining towards him? We will never know.
We should not lose sight of an important conclusion: according to this mathematical model, the timings slightly favour Parry making the call. However, this is one factor among many that we should consider when trying to infer the identity of the infamous and elusive Mr Qualtrough.
Antony M. Brown, January 2019.
P.S. A thought occurred to me while writing this article. During his second police interview, a guilty Wallace could have safely said he boarded the tram by the phone box. In 1931, a call could not be traced, only logged by the exchange if there was a complaint, and the police seemed unaware of even this. Instead, Wallace claimed he boarded some 500 yards away, by Belmont Road, an unnecessary lie that could have sent him to the gallows had it been discovered. Is this an indication he told the truth?
 In her police statement, Alfreds said the call came through at "about 7.15 pm" - it was not logged. She said the second call came through (to her colleague sitting next to her) two minutes later. If correct, the caller let the phone ring for almost two minutes before re-connecting to the operator, and it took a further three minutes to place the call. Also, if Alfreds was correct, a guilty Wallace must have left his house at 7.11 pm at the latest.
 The nearest tram stop for Wallace was at the end of Richmond Park. Therefore, a stakeout somewhere along this road was a sensible location for Parry, especially as it would also enable a close view of Wallace to ensure identification. However, he would have failed to observe Wallace had he left by the front door, but Parry could not be certain whether Wallace would even leave for the chess club that night. It was as good a place as any, in my view.