Inverness Caledonian Thistle meet Hearts in the other tie, a repeat of last year's last-four game that the Edinburgh side won on penalties. The ties will be played on February 1 and 2, with venues still to be decided. Hampden Park will be out of commission due to preparations for next summer's Commonwealth Games.
On his first day in the job, having succeeded Steve Lomas, Wright was asked by chairman Steve Brown to win St Johnstone's their first major piece of silverware in their 129-year history. He concedes that it would be a remarkable achievement if they could finally break that duck. "I want to get the monkey off our backs of not having won a League Cup or Scottish Cup," he said. "How do you measure success at St Johnstone? Finishing in the top six is one way, another is to win a trophy.
"I sat down with the chairman and that's the first thing he said: 'I want you to win me a cup'. No pressure. The club is almost 130 years old and have never won one but I'm the one to do it? If you did you'd take a lot of satisfaction but there's a long way to go. As I said to Callum Davidson [his assistant] with the semi-final being in February at least we should still be in a job by then."
Wright hopes St Johnstone can take heart from the successes of Kilmarnock and St Mirren, the last two clubs to lift the League Cup. "Celtic were beaten in the final by Kilmarnock and in the semi-final by St Mirren," he said. "Even down in England, the Capital One Cup has thrown up smaller clubs winning it. With us coming in later this year [having been in Europe] we've only got four matches to win it. It does give you a great opportunity. We've got to put a huge emphasis on the cup competitions; we'll never treat them lightly. It's a great opportunity for the club to get silverware."
With Aberdeen likely to be backed by a large travelling support, Wright hopes the semi-final will be played at a suitable arena. "For atmosphere we're probably better at one of the Edinburgh grounds or Tannadice," he said. "It'll be determined on how many fans Aberdeen feel they will bring. They brought 17,000 to a semi-final at Hampden before, so if they turn around and say they can get 20,000 I don't know where they'd take it. It's probably between Easter Road, Tynecastle and Tannadice. As a player it didn't matter if the opposition had 40,000, I revelled in it. I don't think it'd be intimidating, it'd just be a good atmosphere."
As well as the crowd, St Johnstone will have to overcome their history. Derek McInnes, now the manager of Aberdeen, took the Perth side to their most recent semi-final, in the Scottish Cup in 2008, but that was one of three fruitless visits they have made to the last four of a major cup competitions in the last six years. Still, that statistic does at least afford them a significantly superior cup record than Gary McDonald, the man whose goal against Morton earned the last-four place against one of his former clubs.
"This is the first time for me in a semi-final so it has not been a great record," said McDonald, who was at Aberdeen between 2008 and 2010. "We want to go all the way now. It's a great chance to make history; a fantastic opportunity now that we're in the semi-final and favourites Celtic have gone. There are four Premiership teams involved and it's anybody's. It's who plays best on the day."
That being so, St Johnstone will need to play much better than they did against Morton, bottom of the SPFL Championship, on Wednesday. While McDonald deserves credit for having the mixture of work ethic and optimism to be in position to profit from Nicolas Caraux's error from Stevie May's controversially retaken 92nd-minute free-kick, the Perth side prospered despite playing poorly.
"I didn't see it but the ref marked out 10 yards before the free-kick was taken and he knew where the marker was," said McDonald. "They're upset as it was so late in the game and they played well. We'll have to wait and see but after a match like that you do wonder if it is your year. We managed to get through and hopefully it is. But we've got to make sure we play better in the semi-final than we did against Morton."