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 1 time and lost 9 times.

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

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 348 times and lost 652 times.

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

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 88 times and lost 912 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 37 times and lost 63 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;
	}
}