The plot of this episode revolves around a cleverly engineered misunderstanding that Lucy and Ethel have with a woman (Edna Grundy) they are sharing a ride with down to Florida. While making a stop for Edna to nap, Lucy and Ethel hear on the car radio that the search is on for a hatchet murderess. By coincidence, they had just discovered a hatchet in the trunk of Edna's car prior to hearing the announcement and jump to the conclusion that Edna is the murderess. They continue their ride with the two struggling to stay awake as Edna drives--but unfortunately they doze off just before Edna switches on the radio to hear that the hatchet murderess recently dyed her hair red and was last seen with a blonde accomplice.
Though the plot seems highly improbable, the comedy works because the misunderstandings are understandable. We can see why Lucy and Ethel would jump to conclusions after riding with a driver who not only has a hatchet in her trunk, but seems to be in a hurry to get to Florida and prefers to take the back roads. Also, Lucy and Ethel dozing off during the car ride an experience that any one who has taken a long road trip can relate to. The end result is two parties riding along in the same car, each one thinking the other is a murderer. Great use of dramatic irony to turn an unlikely situation into a hilarious one!