Sadly Stoke's away record is the worst in the division - there has been just one win, back in October, and a total haul of a meagre eight points.
Things are pretty rosy for West Ham at the moment after their point at Chelsea gave them a bit of breathing space above the bottom three. If we did, the football gods really would be against us'.
Hart's failure to impress came while his Stoke counterpart Jack Butland - another contender for the England number one shirt - produced a solid display.
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Marko Arnautovic and Javier Hernandez will be hoping to continue their goal-scoring form; scoring three in their last two games between the pair.
Hart spilled a shot from Xherdan Shaqiri that came in from the top of the box, while the veteran Englishman was properly positioned to make West Ham pay for the error.
Stoke was close to a huge win in its fight to avoid the drop when Carroll, who had been on the field as a substitute for just four minutes, met a left-wing cross with a crisp volley into the bottom corner.
"He's someone who needs the confidence, but I think he also needs the understanding of how you get to the levels you've got to".
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The former Liverpool striker's first goal since November put relegation-threatened Stoke within touching distance of their first win in 10 games.
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Stoke City have not won a game since 20 January this year - a 2-0 win over Huddersfield Town.
Coverage of today's Premier League game between West Ham and Stoke City will be shown live on Sky Sports Main Event (Sky channel 401, Sky HD 435, Virgin channel 511) and Sky Sports Premier League (Sky channel 402, Sky HD 436, Virgin channel 512). No one is seriously asking Gareth Southgate to include Andy Carroll or Peter Crouch in his World Cup squad and yet is there anything more traditionally English than the sight of both of them giving their struggling teams a lifeline?
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Only Stoke (63) and Watford (60) have conceded more league goals this than West Ham (58) this season.