Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 3 times and lost 7 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Car
6. Car
7. Goat
8. Goat
9. Goat
10. Car

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 331 times and lost 669 times.

1. Car
2. Goat
3. Car
4. Goat
5. Car
6. Goat
7. Goat
8. Goat
9. Car
10. Car
11. Car
12. Goat
13. Goat
14. Goat
15. Goat
16. Goat
17. Car
18. Goat
19. Goat
20. Goat
21. Goat
22. Goat
23. Goat
24. Car
25. Goat
26. Car
27. Goat
28. Goat
29. Goat
30. Car
31. Car
32. Goat
33. Goat
34. Goat
35. Goat
36. Goat
37. Goat
38. Goat
39. Car
40. Car
41. Goat
42. Goat
43. Car
44. Goat
45. Goat
46. Goat
47. Car
48. Goat
49. Goat
50. Car
51. Goat
52. Goat
53. Car
54. Goat
55. Car
56. Goat
57. Car
58. Goat
59. Goat
60. Goat
61. Goat
62. Goat
63. Goat
64. Car
65. Goat
66. Goat
67. Goat
68. Goat
69. Goat
70. Car
71. Goat
72. Goat
73. Goat
74. Car
75. Goat
76. Goat
77. Goat
78. Goat
79. Goat
80. Car
81. Goat
82. Goat
83. Goat
84. Goat
85. Car
86. Goat
87. Car
88. Goat
89. Goat
90. Car
91. Goat
92. Car
93. Goat
94. Car
95. Car
96. Goat
97. Goat
98. Car
99. Goat
100. Goat
101. Goat
102. Goat
103. Car
104. Car
105. Goat
106. Goat
107. Car
108. Goat
109. Goat
110. Goat
111. Car
112. Goat
113. Goat
114. Goat
115. Car
116. Car
117. Goat
118. Goat
119. Car
120. Goat
121. Goat
122. Goat
123. Goat
124. Goat
125. Goat
126. Car
127. Car
128. Goat
129. Goat
130. Goat
131. Car
132. Car
133. Car
134. Goat
135. Goat
136. Car
137. Goat
138. Goat
139. Goat
140. Goat
141. Goat
142. Car
143. Goat
144. Car
145. Goat
146. Goat
147. Car
148. Car
149. Car
150. Goat
151. Car
152. Goat
153. Car
154. Goat
155. Goat
156. Car
157. Car
158. Car
159. Goat
160. Goat
161. Car
162. Goat
163. Car
164. Goat
165. Goat
166. Goat
167. Car
168. Goat
169. Goat
170. Car
171. Car
172. Goat
173. Goat
174. Car
175. Goat
176. Goat
177. Goat
178. Goat
179. Car
180. Car
181. Goat
182. Goat
183. Goat
184. Goat
185. Goat
186. Car
187. Car
188. Goat
189. Goat
190. Goat
191. Goat
192. Goat
193. Car
194. Goat
195. Goat
196. Car
197. Car
198. Car
199. Goat
200. Car
201. Goat
202. Car
203. Goat
204. Goat
205. Goat
206. Goat
207. Goat
208. Goat
209. Goat
210. Car
211. Goat
212. Goat
213. Car
214. Goat
215. Car
216. Car
217. Goat
218. Goat
219. Goat
220. Goat
221. Car
222. Car
223. Goat
224. Goat
225. Goat
226. Goat
227. Goat
228. Goat
229. Car
230. Goat
231. Car
232. Goat
233. Car
234. Goat
235. Car
236. Goat
237. Car
238. Goat
239. Goat
240. Goat
241. Goat
242. Goat
243. Car
244. Goat
245. Goat
246. Car
247. Goat
248. Goat
249. Car
250. Goat
251. Goat
252. Goat
253. Goat
254. Goat
255. Goat
256. Car
257. Car
258. Goat
259. Goat
260. Car
261. Goat
262. Car
263. Goat
264. Car
265. Car
266. Goat
267. Goat
268. Goat
269. Goat
270. Goat
271. Car
272. Goat
273. Car
274. Car
275. Goat
276. Goat
277. Goat
278. Car
279. Goat
280. Goat
281. Goat
282. Goat
283. Goat
284. Car
285. Car
286. Goat
287. Goat
288. Car
289. Goat
290. Goat
291. Goat
292. Car
293. Goat
294. Car
295. Goat
296. Goat
297. Goat
298. Car
299. Goat
300. Car
301. Goat
302. Goat
303. Goat
304. Car
305. Car
306. Car
307. Goat
308. Car
309. Car
310. Car
311. Goat
312. Goat
313. Car
314. Car
315. Goat
316. Car
317. Car
318. Car
319. Goat
320. Car
321. Goat
322. Car
323. Car
324. Goat
325. Goat
326. Goat
327. Goat
328. Goat
329. Goat
330. Goat
331. Car
332. Goat
333. Goat
334. Goat
335. Goat
336. Goat
337. Goat
338. Car
339. Goat
340. Goat
341. Car
342. Goat
343. Goat
344. Goat
345. Goat
346. Car
347. Car
348. Goat
349. Goat
350. Car
351. Goat
352. Goat
353. Car
354. Goat
355. Goat
356. Goat
357. Car
358. Car
359. Goat
360. Goat
361. Car
362. Goat
363. Car
364. Car
365. Goat
366. Goat
367. Goat
368. Goat
369. Goat
370. Goat
371. Car
372. Goat
373. Car
374. Car
375. Car
376. Car
377. Car
378. Goat
379. Goat
380. Goat
381. Goat
382. Goat
383. Goat
384. Goat
385. Car
386. Goat
387. Car
388. Goat
389. Goat
390. Goat
391. Goat
392. Goat
393. Car
394. Goat
395. Goat
396. Car
397. Goat
398. Goat
399. Goat
400. Goat
401. Car
402. Goat
403. Goat
404. Goat
405. Goat
406. Car
407. Goat
408. Goat
409. Car
410. Goat
411. Goat
412. Goat
413. Goat
414. Car
415. Car
416. Car
417. Goat
418. Goat
419. Goat
420. Goat
421. Goat
422. Goat
423. Goat
424. Goat
425. Goat
426. Car
427. Car
428. Goat
429. Goat
430. Car
431. Car
432. Car
433. Goat
434. Goat
435. Car
436. Goat
437. Car
438. Car
439. Car
440. Goat
441. Goat
442. Goat
443. Car
444. Car
445. Car
446. Car
447. Goat
448. Goat
449. Car
450. Goat
451. Car
452. Car
453. Car
454. Goat
455. Goat
456. Goat
457. Car
458. Car
459. Goat
460. Goat
461. Goat
462. Goat
463. Car
464. Goat
465. Car
466. Car
467. Goat
468. Goat
469. Car
470. Goat
471. Goat
472. Goat
473. Goat
474. Car
475. Goat
476. Goat
477. Goat
478. Goat
479. Goat
480. Goat
481. Car
482. Car
483. Car
484. Goat
485. Goat
486. Goat
487. Goat
488. Goat
489. Goat
490. Goat
491. Car
492. Goat
493. Car
494. Goat
495. Car
496. Goat
497. Car
498. Car
499. Car
500. Car
501. Goat
502. Goat
503. Goat
504. Goat
505. Car
506. Goat
507. Goat
508. Goat
509. Goat
510. Car
511. Goat
512. Car
513. Goat
514. Goat
515. Goat
516. Goat
517. Goat
518. Car
519. Goat
520. Car
521. Car
522. Goat
523. Goat
524. Car
525. Goat
526. Car
527. Goat
528. Car
529. Car
530. Car
531. Goat
532. Goat
533. Goat
534. Goat
535. Car
536. Goat
537. Goat
538. Car
539. Goat
540. Car
541. Goat
542. Car
543. Car
544. Goat
545. Goat
546. Goat
547. Car
548. Goat
549. Car
550. Goat
551. Car
552. Goat
553. Goat
554. Goat
555. Goat
556. Goat
557. Goat
558. Car
559. Goat
560. Goat
561. Car
562. Goat
563. Goat
564. Goat
565. Goat
566. Goat
567. Goat
568. Car
569. Goat
570. Goat
571. Goat
572. Goat
573. Car
574. Goat
575. Goat
576. Goat
577. Goat
578. Goat
579. Goat
580. Goat
581. Goat
582. Goat
583. Goat
584. Goat
585. Goat
586. Goat
587. Goat
588. Goat
589. Goat
590. Goat
591. Goat
592. Car
593. Car
594. Car
595. Goat
596. Car
597. Goat
598. Goat
599. Car
600. Goat
601. Car
602. Car
603. Goat
604. Car
605. Goat
606. Goat
607. Goat
608. Car
609. Goat
610. Car
611. Goat
612. Goat
613. Goat
614. Goat
615. Goat
616. Goat
617. Goat
618. Car
619. Car
620. Goat
621. Goat
622. Car
623. Goat
624. Goat
625. Car
626. Goat
627. Goat
628. Car
629. Car
630. Goat
631. Goat
632. Goat
633. Goat
634. Goat
635. Goat
636. Car
637. Goat
638. Goat
639. Car
640. Car
641. Goat
642. Goat
643. Goat
644. Goat
645. Car
646. Car
647. Goat
648. Goat
649. Goat
650. Car
651. Goat
652. Goat
653. Car
654. Goat
655. Goat
656. Goat
657. Goat
658. Goat
659. Goat
660. Car
661. Car
662. Goat
663. Goat
664. Goat
665. Car
666. Goat
667. Goat
668. Goat
669. Goat
670. Goat
671. Goat
672. Goat
673. Goat
674. Car
675. Goat
676. Car
677. Goat
678. Car
679. Goat
680. Goat
681. Goat
682. Goat
683. Goat
684. Goat
685. Goat
686. Car
687. Goat
688. Goat
689. Car
690. Goat
691. Goat
692. Goat
693. Goat
694. Goat
695. Goat
696. Goat
697. Car
698. Car
699. Goat
700. Goat
701. Car
702. Goat
703. Goat
704. Car
705. Goat
706. Goat
707. Car
708. Car
709. Goat
710. Goat
711. Goat
712. Goat
713. Goat
714. Goat
715. Goat
716. Car
717. Goat
718. Goat
719. Car
720. Goat
721. Goat
722. Goat
723. Goat
724. Car
725. Goat
726. Car
727. Goat
728. Car
729. Goat
730. Goat
731. Goat
732. Goat
733. Goat
734. Goat
735. Goat
736. Goat
737. Goat
738. Car
739. Goat
740. Goat
741. Goat
742. Goat
743. Goat
744. Goat
745. Car
746. Goat
747. Goat
748. Goat
749. Car
750. Car
751. Goat
752. Car
753. Goat
754. Goat
755. Car
756. Car
757. Goat
758. Goat
759. Car
760. Goat
761. Goat
762. Car
763. Car
764. Goat
765. Car
766. Car
767. Goat
768. Goat
769. Car
770. Goat
771. Goat
772. Car
773. Goat
774. Goat
775. Goat
776. Car
777. Goat
778. Car
779. Car
780. Car
781. Goat
782. Car
783. Goat
784. Goat
785. Car
786. Goat
787. Goat
788. Car
789. Goat
790. Goat
791. Goat
792. Goat
793. Goat
794. Goat
795. Car
796. Goat
797. Goat
798. Goat
799. Car
800. Goat
801. Goat
802. Goat
803. Goat
804. Goat
805. Car
806. Goat
807. Goat
808. Goat
809. Car
810. Goat
811. Car
812. Goat
813. Goat
814. Car
815. Goat
816. Goat
817. Goat
818. Goat
819. Goat
820. Car
821. Goat
822. Goat
823. Goat
824. Goat
825. Goat
826. Car
827. Car
828. Car
829. Goat
830. Car
831. Goat
832. Car
833. Goat
834. Goat
835. Car
836. Goat
837. Car
838. Goat
839. Car
840. Car
841. Goat
842. Car
843. Goat
844. Goat
845. Car
846. Goat
847. Goat
848. Goat
849. Goat
850. Goat
851. Goat
852. Car
853. Goat
854. Goat
855. Goat
856. Goat
857. Goat
858. Car
859. Goat
860. Car
861. Car
862. Goat
863. Car
864. Goat
865. Goat
866. Goat
867. Car
868. Goat
869. Goat
870. Goat
871. Goat
872. Goat
873. Car
874. Goat
875. Goat
876. Goat
877. Car
878. Goat
879. Goat
880. Goat
881. Goat
882. Car
883. Car
884. Car
885. Goat
886. Goat
887. Car
888. Goat
889. Car
890. Goat
891. Goat
892. Car
893. Goat
894. Goat
895. Car
896. Goat
897. Goat
898. Goat
899. Goat
900. Goat
901. Goat
902. Goat
903. Car
904. Car
905. Car
906. Goat
907. Goat
908. Goat
909. Goat
910. Goat
911. Goat
912. Goat
913. Goat
914. Goat
915. Goat
916. Goat
917. Goat
918. Goat
919. Goat
920. Car
921. Goat
922. Goat
923. Goat
924. Goat
925. Car
926. Car
927. Goat
928. Goat
929. Goat
930. Car
931. Car
932. Car
933. Goat
934. Goat
935. Goat
936. Goat
937. Car
938. Goat
939. Car
940. Car
941. Goat
942. Goat
943. Car
944. Goat
945. Goat
946. Car
947. Car
948. Goat
949. Goat
950. Car
951. Goat
952. Goat
953. Goat
954. Car
955. Car
956. Goat
957. Goat
958. Goat
959. Goat
960. Car
961. Goat
962. Goat
963. Goat
964. Car
965. Goat
966. Goat
967. Car
968. Car
969. Car
970. Car
971. Goat
972. Car
973. Goat
974. Goat
975. Goat
976. Goat
977. Car
978. Car
979. Goat
980. Goat
981. Car
982. Goat
983. Goat
984. Goat
985. Car
986. Goat
987. Car
988. Car
989. Car
990. Car
991. Car
992. Goat
993. Goat
994. Goat
995. Car
996. Car
997. Goat
998. Goat
999. Goat
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 85 times and lost 915 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 33 times and lost 67 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}